Formulas

Arithmetic and Ratios

35 formulas

0001 / Arithmetic and Ratios

Percent calculations: part from base and percent

\[Part = Base \cdot \frac{Percent}{100}\]

These formulas convert between a part, a base amount, and a percentage.

0002 / Arithmetic and Ratios

Percent calculations: percent from part and base

\[Percent = \frac{Part}{Base} \cdot 100\]

These formulas convert between a part, a base amount, and a percentage.

0003 / Arithmetic and Ratios

Percent calculations: base from part and percent

\[Base = \frac{Part}{Percent / 100}\]

These formulas convert between a part, a base amount, and a percentage.

0004 / Arithmetic and Ratios

Percent change: percent change

\[\%\ Change = \frac{New - Old}{Old} \cdot 100\]

Percent change compares a new value to an original value.

0005 / Arithmetic and Ratios

Percent change: new value from original and percent change

\[New = Old \left(1 + \frac{\%\ Change}{100}\right)\]

Percent change compares a new value to an original value.

0006 / Arithmetic and Ratios

Percent change: original value from new value and percent change

\[Old = \frac{New}{1 + \frac{\%\ Change}{100}}\]

Percent change compares a new value to an original value.

0007 / Arithmetic and Ratios

Ratio and proportion: ratio

\[Ratio = \frac{a}{b}\]

Ratios compare quantities and proportions set two ratios equal.

0008 / Arithmetic and Ratios

Ratio and proportion: proportion cross product

\[ad = bc\]

If \(\frac{a}{b} = \frac{c}{d}\), then cross multiplication gives \(ad = bc\).

0009 / Arithmetic and Ratios

Ratio and proportion: unit rate

\[Unit\ Rate = \frac{Quantity}{Units}\]

Ratios compare quantities and proportions set two ratios equal.

0010 / Arithmetic and Ratios

Arithmetic mean

\[\bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n}\]

The arithmetic mean adds all observations and divides by the count.

0011 / Arithmetic and Ratios

Weighted mean

\[\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}\]

A weighted average gives each value a specified weight.

0012 / Arithmetic and Ratios

Geometric mean

\[GM = \sqrt[n]{x_1 x_2 \cdots x_n}\]

The geometric mean is common in growth-rate and ratio problems.

0013 / Arithmetic and Ratios

Harmonic mean

\[HM = \frac{n}{\sum \frac{1}{x_i}}\]

The harmonic mean is useful for rates such as speed over equal distances.

0014 / Arithmetic and Ratios

Distance from rate and time

\[d = rt\]

Distance equals rate multiplied by time.

0015 / Arithmetic and Ratios

Rate from distance and time

\[r = \frac{d}{t}\]

Rate is distance divided by time.

0016 / Arithmetic and Ratios

Time from distance and rate

\[t = \frac{d}{r}\]

Time is distance divided by rate.

0017 / Arithmetic and Ratios

Combined work rate

\[\frac{1}{T} = \frac{1}{T_1} + \frac{1}{T_2}\]

For two workers, combined work rate is the sum of individual rates.

0018 / Arithmetic and Ratios

Work rate

\[Rate = \frac{Work}{Time}\]

Work rate measures how much work is completed per unit of time.

0019 / Arithmetic and Ratios

Density

\[\rho = \frac{m}{V}\]

Density is mass divided by volume.

0020 / Arithmetic and Ratios

Mass from density

\[m = \rho V\]

Mass equals density times volume.

0021 / Arithmetic and Ratios

Volume from density

\[V = \frac{m}{\rho}\]

Volume equals mass divided by density.

0022 / Arithmetic and Ratios

Concentration

\[C = \frac{Amount\ of\ Solute}{Total\ Solution}\]

Concentration compares dissolved material to total solution.

0023 / Arithmetic and Ratios

Dilution

\[C_1V_1 = C_2V_2\]

Dilution keeps the amount of solute constant before and after mixing.

0024 / Arithmetic and Ratios

Markup from cost

\[Selling\ Price = Cost \left(1 + \frac{Markup}{100}\right)\]

Markup adds a percent of cost to the original cost.

0025 / Arithmetic and Ratios

Markdown from list price

\[Sale\ Price = List\ Price \left(1 - \frac{Markdown}{100}\right)\]

Markdown subtracts a percentage from list price.

0026 / Arithmetic and Ratios

Sales tax added

\[Total = Price \left(1 + \frac{Tax}{100}\right)\]

Sales tax increases the pretax price by the tax rate.

0027 / Arithmetic and Ratios

Pretax price from total

\[Price = \frac{Total}{1 + \frac{Tax}{100}}\]

Divide the final total by one plus the tax rate.

0028 / Arithmetic and Ratios

Commission

\[Commission = Sales \cdot \frac{Rate}{100}\]

Commission pay is a percentage of sales.

0029 / Arithmetic and Ratios

Tip amount

\[Tip = Bill \cdot \frac{Rate}{100}\]

Tip is computed as a percentage of the bill.

0030 / Arithmetic and Ratios

Final bill with tip

\[Total = Bill + Tip\]

The total owed is the bill plus the tip amount.

0031 / Arithmetic and Ratios

Simple discount

\[Discount = Price \cdot \frac{Rate}{100}\]

A simple discount removes a percentage of the original price.

0032 / Arithmetic and Ratios

Net price after discount

\[Net\ Price = Price - Discount\]

Subtract the discount from the original price.

0033 / Arithmetic and Ratios

Scale factor

\[Scale\ Factor = \frac{New\ Length}{Original\ Length}\]

A scale factor compares a new measurement to the original.

0034 / Arithmetic and Ratios

Direct variation

\[y = kx\]

In direct variation, the ratio \(\frac{y}{x}\) stays constant.

0035 / Arithmetic and Ratios

Inverse variation

\[y = \frac{k}{x}\]

In inverse variation, the product \(xy\) stays constant.

Algebra and Functions

50 formulas

0036 / Algebra and Functions

Linear equation

\[ax + b = c\]

A linear equation has degree one and graphs as a straight line.

0037 / Algebra and Functions

Solve a linear equation

\[x = \frac{c - b}{a}\]

Subtract the constant term and divide by the coefficient of x.

0038 / Algebra and Functions

Slope from two points

\[m = \frac{y_2 - y_1}{x_2 - x_1}\]

Slope measures the change in y over the change in x.

0039 / Algebra and Functions

Point-slope form

\[y - y_1 = m(x - x_1)\]

Point-slope form builds a line from one point and a slope.

0040 / Algebra and Functions

Slope-intercept form

\[y = mx + b\]

Slope-intercept form shows the slope and y-intercept directly.

0041 / Algebra and Functions

Standard line form

\[Ax + By = C\]

Standard form is common for systems of equations.

0042 / Algebra and Functions

Midpoint formula

\[\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\]

The midpoint lies halfway between two points.

0043 / Algebra and Functions

Distance formula

\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]

The distance formula comes from the Pythagorean theorem.

0044 / Algebra and Functions

Section formula

\[\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)\]

This gives the point dividing a segment in the ratio m:n.

0045 / Algebra and Functions

Quadratic formula

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

The quadratic formula solves \(ax^2 + bx + c = 0\).

0046 / Algebra and Functions

Quadratic discriminant

\[\Delta = b^2 - 4ac\]

The discriminant tells how many real roots a quadratic has.

0047 / Algebra and Functions

Vertex x-coordinate

\[x_v = -\frac{b}{2a}\]

The axis of symmetry of a parabola passes through the vertex.

0048 / Algebra and Functions

Vertex y-coordinate

\[y_v = f\left(-\frac{b}{2a}\right)\]

Substitute the vertex x-value into the quadratic to get y.

0049 / Algebra and Functions

Sum of quadratic roots

\[r_1 + r_2 = -\frac{b}{a}\]

Vieta's formulas connect coefficients to roots.

0050 / Algebra and Functions

Product of quadratic roots

\[r_1 r_2 = \frac{c}{a}\]

The product of roots follows from Vieta's formulas.

0051 / Algebra and Functions

Difference of squares

\[a^2 - b^2 = (a-b)(a+b)\]

A difference of squares factors into conjugates.

0052 / Algebra and Functions

Perfect square trinomial

\[a^2 + 2ab + b^2 = (a+b)^2\]

Use this identity when a trinomial matches a square pattern.

0053 / Algebra and Functions

Perfect square trinomial with subtraction

\[a^2 - 2ab + b^2 = (a-b)^2\]

This identity expands a squared difference.

0054 / Algebra and Functions

Binomial cube sum

\[(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\]

This is the standard expansion for a binomial cube.

0055 / Algebra and Functions

Binomial cube difference

\[(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\]

Use alternating signs for the cube of a difference.

0056 / Algebra and Functions

Sum of cubes

\[a^3 + b^3 = (a+b)(a^2 - ab + b^2)\]

This factors a sum of two cubes.

0057 / Algebra and Functions

Difference of cubes

\[a^3 - b^3 = (a-b)(a^2 + ab + b^2)\]

This factors a difference of two cubes.

0058 / Algebra and Functions

Exponent product rule

\[a^m a^n = a^{m+n}\]

When multiplying like bases, add exponents.

0059 / Algebra and Functions

Exponent quotient rule

\[\frac{a^m}{a^n} = a^{m-n}\]

When dividing like bases, subtract exponents.

0060 / Algebra and Functions

Power of a power

\[\left(a^m\right)^n = a^{mn}\]

Multiply exponents when raising a power to a power.

0061 / Algebra and Functions

Power of a product

\[(ab)^n = a^n b^n\]

Distribute the exponent across factors.

0062 / Algebra and Functions

Power of a quotient

\[\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\]

Distribute the exponent across numerator and denominator.

0063 / Algebra and Functions

Negative exponent

\[a^{-n} = \frac{1}{a^n}\]

A negative exponent moves the base to the denominator.

0064 / Algebra and Functions

Zero exponent

\[a^0 = 1\]

Any nonzero base raised to the zero power equals one.

0065 / Algebra and Functions

Fractional exponent

\[a^{1/n} = \sqrt[n]{a}\]

A denominator in the exponent corresponds to a root.

0066 / Algebra and Functions

General radical exponent

\[a^{m/n} = \sqrt[n]{a^m}\]

A rational exponent combines powers and roots.

0067 / Algebra and Functions

Common logarithm definition

\[\log_{10}(x) = y \iff 10^y = x\]

A logarithm asks what exponent produces a value.

0068 / Algebra and Functions

Natural logarithm definition

\[\ln(x) = y \iff e^y = x\]

The natural log is the logarithm base e.

0069 / Algebra and Functions

Log product rule

\[\log_b(xy) = \log_b x + \log_b y\]

The log of a product becomes a sum.

0070 / Algebra and Functions

Log quotient rule

\[\log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y\]

The log of a quotient becomes a difference.

0071 / Algebra and Functions

Log power rule

\[\log_b(x^n) = n \log_b x\]

Bring exponents down as coefficients.

0072 / Algebra and Functions

Change of base

\[\log_b x = \frac{\log_k x}{\log_k b}\]

Change-of-base lets you rewrite any logarithm using a convenient base.

0073 / Algebra and Functions

Inverse of an exponential

\[x = \log_b y \iff y = b^x\]

Exponential and logarithmic functions undo each other.

0074 / Algebra and Functions

Arithmetic sequence nth term

\[a_n = a_1 + (n-1)d\]

Arithmetic sequences change by a constant difference.

0075 / Algebra and Functions

Arithmetic series sum

\[S_n = \frac{n}{2}(a_1 + a_n)\]

Sum the first n terms of an arithmetic sequence.

0076 / Algebra and Functions

Geometric sequence nth term

\[a_n = a_1 r^{n-1}\]

Geometric sequences change by a constant ratio.

0077 / Algebra and Functions

Finite geometric series sum

\[S_n = a_1 \frac{1-r^n}{1-r}\]

Use this when the ratio is not one.

0078 / Algebra and Functions

Infinite geometric series sum

\[S_\infty = \frac{a_1}{1-r}\]

This converges only when \(|r| < 1\).

0079 / Algebra and Functions

Function composition

\[(f \circ g)(x) = f(g(x))\]

Composition applies one function inside another.

0080 / Algebra and Functions

Inverse function check

\[f\left(f^{-1}(x)\right) = x\]

An inverse function undoes the original function.

0081 / Algebra and Functions

Average rate of change

\[\frac{f(b) - f(a)}{b-a}\]

This measures how fast a function changes across an interval.

0082 / Algebra and Functions

Complex modulus

\[|a+bi| = \sqrt{a^2 + b^2}\]

The modulus is the distance from the origin in the complex plane.

0083 / Algebra and Functions

Complex conjugate

\[\overline{a+bi} = a-bi\]

The conjugate flips the sign of the imaginary part.

0084 / Algebra and Functions

Euler form

\[re^{i\theta} = r(\cos\theta + i\sin\theta)\]

Euler form converts between polar and rectangular complex numbers.

0085 / Algebra and Functions

De Moivre's theorem

\[\left(r(\cos\theta + i\sin\theta)\right)^n = r^n(\cos n\theta + i\sin n\theta)\]

Use De Moivre's theorem for powers of complex numbers.

Plane Geometry

63 formulas

0086 / Plane Geometry

Square area

\[A = s^2\]

Area of a square from one side length.

0087 / Plane Geometry

Square perimeter

\[P = 4s\]

Perimeter is four times the side length.

0088 / Plane Geometry

Square diagonal

\[d = s\sqrt{2}\]

The diagonal follows from the Pythagorean theorem.

0089 / Plane Geometry

Square side from area

\[s = \sqrt{A}\]

Take the square root of the area.

0090 / Plane Geometry

Square side from perimeter

\[s = \frac{P}{4}\]

Divide the perimeter by four.

0091 / Plane Geometry

Rectangle area

\[A = lw\]

Rectangle area is length times width.

0092 / Plane Geometry

Rectangle perimeter

\[P = 2l + 2w\]

Add both length and width pairs.

0093 / Plane Geometry

Rectangle diagonal

\[d = \sqrt{l^2 + w^2}\]

Use the Pythagorean theorem on the sides.

0094 / Plane Geometry

Rectangle length from area

\[l = \frac{A}{w}\]

Solve the area formula for length.

0095 / Plane Geometry

Rectangle width from area

\[w = \frac{A}{l}\]

Solve the area formula for width.

0096 / Plane Geometry

Triangle area

\[A = \frac{1}{2}bh\]

Use base times height divided by two.

0097 / Plane Geometry

Triangle perimeter

\[P = a + b + c\]

Perimeter is the sum of side lengths.

0098 / Plane Geometry

Pythagorean theorem

\[a^2 + b^2 = c^2\]

This relates the legs and hypotenuse of a right triangle.

0099 / Plane Geometry

Heron's formula

\[A = \sqrt{s(s-a)(s-b)(s-c)}\]

Heron's formula uses all three side lengths and the semiperimeter.

0100 / Plane Geometry

Triangle semiperimeter

\[s = \frac{a+b+c}{2}\]

The semiperimeter is half the full perimeter.

0101 / Plane Geometry

Triangle inradius

\[r = \frac{A}{s}\]

The inradius equals area divided by semiperimeter.

0102 / Plane Geometry

Triangle circumradius

\[R = \frac{abc}{4A}\]

Use all three sides and the area.

0103 / Plane Geometry

Equilateral triangle area

\[A = \frac{\sqrt{3}}{4}s^2\]

An equilateral triangle has all sides equal.

0104 / Plane Geometry

Equilateral triangle height

\[h = \frac{\sqrt{3}}{2}s\]

The height splits the triangle into two 30-60-90 triangles.

0105 / Plane Geometry

Equilateral triangle perimeter

\[P = 3s\]

Perimeter is three times one side length.

0106 / Plane Geometry

Parallelogram area

\[A = bh\]

Area is base times perpendicular height.

0107 / Plane Geometry

Parallelogram perimeter

\[P = 2(a+b)\]

Opposite sides are equal.

0108 / Plane Geometry

Parallelogram base from area

\[b = \frac{A}{h}\]

Solve the area formula for base.

0109 / Plane Geometry

Parallelogram height from area

\[h = \frac{A}{b}\]

Solve the area formula for height.

0110 / Plane Geometry

Trapezoid area

\[A = \frac{1}{2}(b_1+b_2)h\]

Average the parallel bases and multiply by height.

0111 / Plane Geometry

Trapezoid median

\[m = \frac{b_1+b_2}{2}\]

The median equals the average of the bases.

0112 / Plane Geometry

Trapezoid height from area

\[h = \frac{2A}{b_1+b_2}\]

Solve the area formula for height.

0113 / Plane Geometry

Trapezoid area from median

\[A = mh\]

Area also equals median times height.

0114 / Plane Geometry

Rhombus area from diagonals

\[A = \frac{1}{2}d_1d_2\]

Multiply the diagonals and divide by two.

0115 / Plane Geometry

Rhombus perimeter

\[P = 4s\]

All sides of a rhombus are equal.

0116 / Plane Geometry

Rhombus side from perimeter

\[s = \frac{P}{4}\]

Divide the perimeter by four.

0117 / Plane Geometry

Rhombus diagonal relation

\[s^2 = \left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2\]

The diagonals bisect each other at right angles.

0118 / Plane Geometry

Kite area

\[A = \frac{1}{2}d_1d_2\]

Kite area uses the two diagonals.

0119 / Plane Geometry

Kite perimeter

\[P = 2(a+b)\]

A kite has two pairs of adjacent equal sides.

0120 / Plane Geometry

Kite diagonal from area

\[d_1 = \frac{2A}{d_2}\]

Solve the area formula for a missing diagonal.

0121 / Plane Geometry

Circle circumference

\[C = 2\pi r\]

Circumference measures the distance around a circle.

0122 / Plane Geometry

Circle area

\[A = \pi r^2\]

Area grows with the square of the radius.

0123 / Plane Geometry

Circle diameter

\[d = 2r\]

The diameter is twice the radius.

0124 / Plane Geometry

Circle radius from diameter

\[r = \frac{d}{2}\]

Half the diameter gives the radius.

0125 / Plane Geometry

Circle radius from circumference

\[r = \frac{C}{2\pi}\]

Rearrange the circumference formula.

0126 / Plane Geometry

Circle radius from area

\[r = \sqrt{\frac{A}{\pi}}\]

Rearrange the area formula and take the square root.

0127 / Plane Geometry

Arc length

\[s = r\theta\]

Use radians for the central angle.

0128 / Plane Geometry

Sector area

\[A = \frac{1}{2}r^2\theta\]

Use radians for the central angle.

0129 / Plane Geometry

Chord length

\[c = 2r\sin\left(\frac{\theta}{2}\right)\]

Chord length depends on radius and central angle.

0130 / Plane Geometry

Ellipse area

\[A = \pi ab\]

Multiply the semi-major and semi-minor axes by pi.

0131 / Plane Geometry

Ellipse eccentricity

\[e = \sqrt{1 - \frac{b^2}{a^2}}\]

Eccentricity measures how stretched an ellipse is.

0132 / Plane Geometry

Ellipse focal distance

\[c = \sqrt{a^2 - b^2}\]

The focus distance comes from the axes.

0133 / Plane Geometry

Ellipse circumference approximation

\[C \approx \pi \left[3(a+b) - \sqrt{(3a+b)(a+3b)}\right]\]

Ramanujan's approximation estimates ellipse circumference accurately.

0134 / Plane Geometry

Regular polygon perimeter

\[P = ns\]

Multiply the number of sides by the side length.

0135 / Plane Geometry

Regular polygon interior-angle sum

\[S = (n-2)180^\circ\]

This gives the total of all interior angles.

0136 / Plane Geometry

Regular polygon one interior angle

\[\alpha = \frac{(n-2)180^\circ}{n}\]

Divide the interior-angle sum by the number of sides.

0137 / Plane Geometry

Regular polygon one exterior angle

\[\beta = \frac{360^\circ}{n}\]

Exterior angles of a regular polygon are equal.

0138 / Plane Geometry

Regular polygon diagonals

\[D = \frac{n(n-3)}{2}\]

Count all non-side connections between vertices.

0139 / Plane Geometry

Regular polygon area

\[A = \frac{1}{2}aP\]

Area uses the apothem and perimeter.

0140 / Plane Geometry

Regular polygon side from perimeter

\[s = \frac{P}{n}\]

Divide the perimeter by the number of sides.

0141 / Plane Geometry

Annulus area

\[A = \pi(R^2-r^2)\]

Subtract the inner circle area from the outer circle area.

0142 / Plane Geometry

Annulus outer radius

\[R = \sqrt{\frac{A}{\pi} + r^2}\]

Solve the annulus area formula for the outer radius.

0143 / Plane Geometry

Annulus inner radius

\[r = \sqrt{R^2 - \frac{A}{\pi}}\]

Solve the annulus area formula for the inner radius.

0144 / Plane Geometry

Circle equation

\[(x-h)^2 + (y-k)^2 = r^2\]

This is the standard equation of a circle centered at (h, k).

0145 / Plane Geometry

Ellipse equation

\[\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\]

This is the standard form of an ellipse centered at (h, k).

0146 / Plane Geometry

Parabola equation

\[(x-h)^2 = 4p(y-k)\]

This vertical parabola opens up or down depending on p.

0147 / Plane Geometry

Hyperbola equation

\[\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\]

This hyperbola opens left-right in standard form.

0148 / Plane Geometry

Shoelace area formula

\[A = \frac{1}{2}\left|\sum_{i=1}^{n} x_i y_{i+1} - y_i x_{i+1}\right|\]

Use vertex coordinates in order around the polygon.

Solid Geometry

35 formulas

0149 / Solid Geometry

Cube volume

\[V = s^3\]

Cube volume is the side length cubed.

0150 / Solid Geometry

Cube surface area

\[SA = 6s^2\]

A cube has six square faces.

0151 / Solid Geometry

Cube space diagonal

\[d = s\sqrt{3}\]

The long diagonal connects opposite vertices.

0152 / Solid Geometry

Rectangular prism volume

\[V = lwh\]

Multiply length, width, and height.

0153 / Solid Geometry

Rectangular prism surface area

\[SA = 2(lw + lh + wh)\]

Add the areas of all faces.

0154 / Solid Geometry

Rectangular prism space diagonal

\[d = \sqrt{l^2 + w^2 + h^2}\]

Use the 3D Pythagorean theorem.

0155 / Solid Geometry

Cylinder volume

\[V = \pi r^2 h\]

Use the circular base area times height.

0156 / Solid Geometry

Cylinder lateral area

\[LA = 2\pi rh\]

This measures the side surface only.

0157 / Solid Geometry

Cylinder total surface area

\[SA = 2\pi r(r+h)\]

Add the two circular bases and the lateral area.

0158 / Solid Geometry

Cone volume

\[V = \frac{1}{3}\pi r^2 h\]

A cone has one-third the volume of a cylinder with the same base and height.

0159 / Solid Geometry

Cone slant height

\[\ell = \sqrt{r^2 + h^2}\]

Slant height follows from a right triangle.

0160 / Solid Geometry

Cone lateral area

\[LA = \pi r\ell\]

Use radius times slant height times pi.

0161 / Solid Geometry

Cone total surface area

\[SA = \pi r(r+\ell)\]

Add the base area and lateral area.

0162 / Solid Geometry

Sphere volume

\[V = \frac{4}{3}\pi r^3\]

Sphere volume grows with the cube of the radius.

0163 / Solid Geometry

Sphere surface area

\[SA = 4\pi r^2\]

Sphere area grows with the square of the radius.

0164 / Solid Geometry

Hemisphere volume

\[V = \frac{2}{3}\pi r^3\]

A hemisphere is half a sphere's volume.

0165 / Solid Geometry

Hemisphere curved surface area

\[CSA = 2\pi r^2\]

Curved surface excludes the circular base.

0166 / Solid Geometry

Hemisphere total surface area

\[TSA = 3\pi r^2\]

Add the base circle to the curved area.

0167 / Solid Geometry

Pyramid volume

\[V = \frac{1}{3}Bh\]

Multiply base area by height and divide by three.

0168 / Solid Geometry

Prism volume

\[V = Bh\]

Any prism volume equals base area times height.

0169 / Solid Geometry

Frustum volume

\[V = \frac{1}{3}\pi h(R^2 + Rr + r^2)\]

This is the volume of a frustum of a cone.

0170 / Solid Geometry

Frustum slant height

\[\ell = \sqrt{h^2 + (R-r)^2}\]

The slant height forms a right triangle with the height and radii difference.

0171 / Solid Geometry

Ellipsoid volume

\[V = \frac{4}{3}\pi abc\]

Use the three semi-axis lengths.

0172 / Solid Geometry

Torus volume

\[V = 2\pi^2 Rr^2\]

R is the major radius and r is the minor radius.

0173 / Solid Geometry

Torus surface area

\[SA = 4\pi^2 Rr\]

Multiply the two circle circumferences in the torus construction.

0174 / Solid Geometry

Regular tetrahedron volume

\[V = \frac{s^3}{6\sqrt{2}}\]

This formula uses the edge length of a regular tetrahedron.

0175 / Solid Geometry

Regular tetrahedron surface area

\[SA = \sqrt{3}s^2\]

A regular tetrahedron has four equilateral triangular faces.

0176 / Solid Geometry

Capsule volume

\[V = \pi r^2 h + \frac{4}{3}\pi r^3\]

A capsule combines a cylinder and two hemispheres.

0177 / Solid Geometry

Capsule surface area

\[SA = 2\pi rh + 4\pi r^2\]

Add cylinder lateral area and sphere area.

0178 / Solid Geometry

Octahedron volume

\[V = \frac{\sqrt{2}}{3}s^3\]

This formula uses the edge length of a regular octahedron.

0179 / Solid Geometry

Octahedron surface area

\[SA = 2\sqrt{3}s^2\]

A regular octahedron has eight equilateral triangles.

0180 / Solid Geometry

Icosahedron volume

\[V = \frac{5(3+\sqrt{5})}{12}s^3\]

This is the volume of a regular icosahedron.

0181 / Solid Geometry

Icosahedron surface area

\[SA = 5\sqrt{3}s^2\]

A regular icosahedron has 20 equilateral triangular faces.

0182 / Solid Geometry

Dodecahedron volume

\[V = \frac{15 + 7\sqrt{5}}{4}s^3\]

This is the volume of a regular dodecahedron.

0183 / Solid Geometry

Dodecahedron surface area

\[SA = 3\sqrt{25 + 10\sqrt{5}}\,s^2\]

A regular dodecahedron has 12 regular pentagonal faces.

Trigonometry

43 formulas

0184 / Trigonometry

Sine definition

\[\sin\theta = \frac{Opposite}{Hypotenuse}\]

In a right triangle, sine compares the opposite side to the hypotenuse.

0185 / Trigonometry

Cosine definition

\[\cos\theta = \frac{Adjacent}{Hypotenuse}\]

Cosine compares the adjacent side to the hypotenuse.

0186 / Trigonometry

Tangent definition

\[\tan\theta = \frac{Opposite}{Adjacent}\]

Tangent compares the opposite side to the adjacent side.

0187 / Trigonometry

Cosecant definition

\[\csc\theta = \frac{1}{\sin\theta}\]

Cosecant is the reciprocal of sine.

0188 / Trigonometry

Secant definition

\[\sec\theta = \frac{1}{\cos\theta}\]

Secant is the reciprocal of cosine.

0189 / Trigonometry

Cotangent definition

\[\cot\theta = \frac{1}{\tan\theta}\]

Cotangent is the reciprocal of tangent.

0190 / Trigonometry

Pythagorean identity

\[\sin^2\theta + \cos^2\theta = 1\]

This is the foundational trigonometric identity.

0191 / Trigonometry

Secant-tangent identity

\[1 + \tan^2\theta = \sec^2\theta\]

Rewrite tangent in terms of secant.

0192 / Trigonometry

Cosecant-cotangent identity

\[1 + \cot^2\theta = \csc^2\theta\]

Rewrite cotangent in terms of cosecant.

0193 / Trigonometry

Sine angle sum

\[\sin(\alpha+\beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta\]

Use this to expand the sine of a sum.

0194 / Trigonometry

Sine angle difference

\[\sin(\alpha-\beta) = \sin\alpha\cos\beta - \cos\alpha\sin\beta\]

Use this to expand the sine of a difference.

0195 / Trigonometry

Cosine angle sum

\[\cos(\alpha+\beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta\]

Use this to expand the cosine of a sum.

0196 / Trigonometry

Cosine angle difference

\[\cos(\alpha-\beta) = \cos\alpha\cos\beta + \sin\alpha\sin\beta\]

Use this to expand the cosine of a difference.

0197 / Trigonometry

Tangent angle sum

\[\tan(\alpha+\beta) = \frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}\]

Use this to combine two tangent angles.

0198 / Trigonometry

Tangent angle difference

\[\tan(\alpha-\beta) = \frac{\tan\alpha-\tan\beta}{1+\tan\alpha\tan\beta}\]

Use this to subtract tangent angles.

0199 / Trigonometry

Sine double-angle

\[\sin(2\theta) = 2\sin\theta\cos\theta\]

This is the double-angle identity for sine.

0200 / Trigonometry

Cosine double-angle

\[\cos(2\theta) = \cos^2\theta - \sin^2\theta\]

One form of the cosine double-angle identity.

0201 / Trigonometry

Cosine double-angle from cosine

\[\cos(2\theta) = 2\cos^2\theta - 1\]

Alternate double-angle form using cosine.

0202 / Trigonometry

Cosine double-angle from sine

\[\cos(2\theta) = 1 - 2\sin^2\theta\]

Alternate double-angle form using sine.

0203 / Trigonometry

Tangent double-angle

\[\tan(2\theta) = \frac{2\tan\theta}{1-\tan^2\theta}\]

This is the double-angle identity for tangent.

0204 / Trigonometry

Sine half-angle

\[\sin\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1-\cos\theta}{2}}\]

Choose the sign from the quadrant.

0205 / Trigonometry

Cosine half-angle

\[\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1+\cos\theta}{2}}\]

Choose the sign from the quadrant.

0206 / Trigonometry

Tangent half-angle

\[\tan\left(\frac{\theta}{2}\right) = \frac{\sin\theta}{1+\cos\theta}\]

One common half-angle form for tangent.

0207 / Trigonometry

Law of sines

\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]

Use the law of sines in non-right triangles.

0208 / Trigonometry

Law of cosines

\[c^2 = a^2 + b^2 - 2ab\cos C\]

This generalizes the Pythagorean theorem.

0209 / Trigonometry

Triangle area from two sides and included angle

\[A = \frac{1}{2}ab\sin C\]

Use this when two sides and the included angle are known.

0210 / Trigonometry

Degree to radian conversion

\[Radians = Degrees \cdot \frac{\pi}{180}\]

Convert degrees to radians by multiplying by pi over 180.

0211 / Trigonometry

Radian to degree conversion

\[Degrees = Radians \cdot \frac{180}{\pi}\]

Convert radians to degrees by multiplying by 180 over pi.

0212 / Trigonometry

Cofunction identity for sine

\[\sin(90^\circ-\theta) = \cos\theta\]

Sine and cosine are cofunctions.

0213 / Trigonometry

Cofunction identity for cosine

\[\cos(90^\circ-\theta) = \sin\theta\]

Cosine and sine are cofunctions.

0214 / Trigonometry

Cofunction identity for tangent

\[\tan(90^\circ-\theta) = \cot\theta\]

Tangent and cotangent are cofunctions.

0215 / Trigonometry

Cofunction identity for cotangent

\[\cot(90^\circ-\theta) = \tan\theta\]

Cotangent and tangent are cofunctions.

0216 / Trigonometry

Cofunction identity for secant

\[\sec(90^\circ-\theta) = \csc\theta\]

Secant and cosecant are cofunctions.

0217 / Trigonometry

Cofunction identity for cosecant

\[\csc(90^\circ-\theta) = \sec\theta\]

Cosecant and secant are cofunctions.

0218 / Trigonometry

Product-to-sum for sine-cosine

\[\sin\alpha\cos\beta = \frac{1}{2}[\sin(\alpha+\beta)+\sin(\alpha-\beta)]\]

Convert products of trig functions into sums.

0219 / Trigonometry

Product-to-sum for cosine-cosine

\[\cos\alpha\cos\beta = \frac{1}{2}[\cos(\alpha+\beta)+\cos(\alpha-\beta)]\]

Convert a cosine product into a sum.

0220 / Trigonometry

Product-to-sum for sine-sine

\[\sin\alpha\sin\beta = \frac{1}{2}[\cos(\alpha-\beta)-\cos(\alpha+\beta)]\]

Convert a sine product into a sum.

0221 / Trigonometry

Sum-to-product for sine

\[\sin\alpha + \sin\beta = 2\sin\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right)\]

Convert a sine sum into a product.

0222 / Trigonometry

Sum-to-product for cosine

\[\cos\alpha + \cos\beta = 2\cos\left(\frac{\alpha+\beta}{2}\right)\cos\left(\frac{\alpha-\beta}{2}\right)\]

Convert a cosine sum into a product.

0223 / Trigonometry

Difference-to-product for cosine

\[\cos\alpha - \cos\beta = -2\sin\left(\frac{\alpha+\beta}{2}\right)\sin\left(\frac{\alpha-\beta}{2}\right)\]

Convert a cosine difference into a product.

0224 / Trigonometry

Unit-circle x-coordinate

\[x = \cos\theta\]

On the unit circle, cosine gives the x-coordinate.

0225 / Trigonometry

Unit-circle y-coordinate

\[y = \sin\theta\]

On the unit circle, sine gives the y-coordinate.

0226 / Trigonometry

Reference-angle tangent

\[\tan\theta = \frac{\sin\theta}{\cos\theta}\]

Tangent can be built from sine and cosine.

Calculus

67 formulas

0227 / Calculus

Derivative definition

\[f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}\]

The derivative is the instantaneous rate of change.

0228 / Calculus

Power rule

\[\frac{d}{dx}x^n = nx^{n-1}\]

Differentiate any power of x by bringing down the exponent.

0229 / Calculus

Constant rule

\[\frac{d}{dx}c = 0\]

The derivative of a constant is zero.

0230 / Calculus

Constant multiple rule

\[\frac{d}{dx}[cf(x)] = c f'(x)\]

Constant factors can be pulled outside the derivative.

0231 / Calculus

Sum rule

\[\frac{d}{dx}[f(x)+g(x)] = f'(x)+g'(x)\]

Differentiate each term separately.

0232 / Calculus

Difference rule

\[\frac{d}{dx}[f(x)-g(x)] = f'(x)-g'(x)\]

Differentiate each term separately.

0233 / Calculus

Product rule

\[\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)\]

Use the product rule when multiplying two functions.

0234 / Calculus

Quotient rule

\[\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}\]

Use the quotient rule when dividing functions.

0235 / Calculus

Chain rule

\[\frac{d}{dx}f(g(x)) = f'(g(x))g'(x)\]

Differentiate outer and inner functions in order.

0236 / Calculus

Derivative of e^x

\[\frac{d}{dx}e^x = e^x\]

The exponential function with base e is its own derivative.

0237 / Calculus

Derivative of a^x

\[\frac{d}{dx}a^x = a^x\ln a\]

Differentiate exponential functions with constant base a.

0238 / Calculus

Derivative of ln x

\[\frac{d}{dx}\ln x = \frac{1}{x}\]

The derivative of the natural logarithm is one over x.

0239 / Calculus

Derivative of log base a

\[\frac{d}{dx}\log_a x = \frac{1}{x\ln a}\]

Use the change-of-base identity to differentiate logs of any base.

0240 / Calculus

Derivative of sin x

\[\frac{d}{dx}\sin x = \cos x\]

Sine differentiates to cosine.

0241 / Calculus

Derivative of cos x

\[\frac{d}{dx}\cos x = -\sin x\]

Cosine differentiates to negative sine.

0242 / Calculus

Derivative of tan x

\[\frac{d}{dx}\tan x = \sec^2 x\]

Differentiate tangent to secant squared.

0243 / Calculus

Derivative of cot x

\[\frac{d}{dx}\cot x = -\csc^2 x\]

Differentiate cotangent to negative cosecant squared.

0244 / Calculus

Derivative of sec x

\[\frac{d}{dx}\sec x = \sec x\tan x\]

Differentiate secant to secant times tangent.

0245 / Calculus

Derivative of csc x

\[\frac{d}{dx}\csc x = -\csc x\cot x\]

Differentiate cosecant to negative cosecant times cotangent.

0246 / Calculus

Derivative of arcsin x

\[\frac{d}{dx}\arcsin x = \frac{1}{\sqrt{1-x^2}}\]

This is valid on the interval where arcsine is defined.

0247 / Calculus

Derivative of arccos x

\[\frac{d}{dx}\arccos x = -\frac{1}{\sqrt{1-x^2}}\]

Arccosine differentiates to a negative reciprocal radical.

0248 / Calculus

Derivative of arctan x

\[\frac{d}{dx}\arctan x = \frac{1}{1+x^2}\]

Arctangent differentiates to one over one plus x squared.

0249 / Calculus

Derivative of sinh x

\[\frac{d}{dx}\sinh x = \cosh x\]

Hyperbolic sine differentiates to hyperbolic cosine.

0250 / Calculus

Derivative of cosh x

\[\frac{d}{dx}\cosh x = \sinh x\]

Hyperbolic cosine differentiates to hyperbolic sine.

0251 / Calculus

Derivative of tanh x

\[\frac{d}{dx}\tanh x = \operatorname{sech}^2 x\]

Hyperbolic tangent differentiates to hyperbolic secant squared.

0252 / Calculus

Derivative of sqrt(x)

\[\frac{d}{dx}\sqrt{x} = \frac{1}{2\sqrt{x}}\]

Rewrite square root as x to the one-half power.

0253 / Calculus

Derivative of 1/x

\[\frac{d}{dx}\left(\frac{1}{x}\right) = -\frac{1}{x^2}\]

Rewrite reciprocal as x to the negative first power.

0254 / Calculus

Derivative of sin(ax+b)

\[\frac{d}{dx}\sin(ax+b) = a\cos(ax+b)\]

Apply the chain rule to a linear inner function.

0255 / Calculus

Derivative of cos(ax+b)

\[\frac{d}{dx}\cos(ax+b) = -a\sin(ax+b)\]

Apply the chain rule to a linear inner function.

0256 / Calculus

Derivative of e^{ax}

\[\frac{d}{dx}e^{ax} = ae^{ax}\]

The inner derivative contributes the constant a.

0257 / Calculus

Derivative of ln(ax+b)

\[\frac{d}{dx}\ln(ax+b) = \frac{a}{ax+b}\]

Combine the log derivative with the chain rule.

0258 / Calculus

Generalized power rule

\[\frac{d}{dx}[u(x)]^n = n[u(x)]^{n-1}u'(x)\]

This extends the power rule by combining it with the chain rule.

0259 / Calculus

Derivative of (ax+b)^n

\[\frac{d}{dx}(ax+b)^n = an(ax+b)^{n-1}\]

Differentiate the outer power, then multiply by the inner derivative.

0260 / Calculus

Derivative of a negative power

\[\frac{d}{dx}x^{-n} = -n x^{-n-1}\]

Negative exponents still follow the general power rule.

0261 / Calculus

Derivative of a fractional power

\[\frac{d}{dx}x^{1/n} = \frac{1}{n}x^{\frac{1}{n}-1}\]

Radical functions are powers with fractional exponents.

0262 / Calculus

n-th derivative of x^m

\[\frac{d^n}{dx^n}x^m = \frac{m!}{(m-n)!}x^{m-n}\]

Repeated differentiation multiplies by descending exponents until the power runs down.

0263 / Calculus

Linear approximation

\[L(x) = f(a) + f'(a)(x-a)\]

Linearization uses the tangent line to approximate a function near x = a.

0264 / Calculus

Tangent line equation

\[y - f(a) = f'(a)(x-a)\]

The tangent line uses the function value and slope at the point of tangency.

0265 / Calculus

Normal line equation

\[y - f(a) = -\frac{1}{f'(a)}(x-a)\]

The normal line is perpendicular to the tangent line when the tangent slope is nonzero.

0266 / Calculus

Newton's method

\[x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\]

Newton's method uses tangent lines to improve root estimates iteratively.

0267 / Calculus

Indefinite integral power rule

\[\int x^n\,dx = \frac{x^{n+1}}{n+1} + C\]

Use this for any power except n = -1.

0268 / Calculus

Integral of 1/x

\[\int \frac{1}{x}\,dx = \ln|x| + C\]

The reciprocal function integrates to a logarithm.

0269 / Calculus

Integral of e^x

\[\int e^x\,dx = e^x + C\]

The exponential function with base e is its own integral.

0270 / Calculus

Integral of a^x

\[\int a^x\,dx = \frac{a^x}{\ln a} + C\]

Integrate an exponential with constant base a.

0271 / Calculus

Integral of sin x

\[\int \sin x\,dx = -\cos x + C\]

Sine integrates to negative cosine.

0272 / Calculus

Integral of cos x

\[\int \cos x\,dx = \sin x + C\]

Cosine integrates to sine.

0273 / Calculus

Integral of sec^2 x

\[\int \sec^2 x\,dx = \tan x + C\]

This is the antiderivative of secant squared.

0274 / Calculus

Integral of csc^2 x

\[\int \csc^2 x\,dx = -\cot x + C\]

This is the antiderivative of cosecant squared.

0275 / Calculus

Integral of sec x tan x

\[\int \sec x\tan x\,dx = \sec x + C\]

Use this when secant times tangent appears together.

0276 / Calculus

Integral of csc x cot x

\[\int \csc x\cot x\,dx = -\csc x + C\]

Use this when cosecant times cotangent appears together.

0277 / Calculus

Integral of 1/(1+x^2)

\[\int \frac{1}{1+x^2}\,dx = \arctan x + C\]

This is the standard inverse-tangent antiderivative.

0278 / Calculus

Integral of 1/sqrt(1-x^2)

\[\int \frac{1}{\sqrt{1-x^2}}\,dx = \arcsin x + C\]

This is the standard inverse-sine antiderivative.

0279 / Calculus

Integration by parts

\[\int u\,dv = uv - \int v\,du\]

Choose u and dv to simplify the integral.

0280 / Calculus

u-substitution

\[\int f(g(x))g'(x)\,dx = \int f(u)\,du\]

Use substitution when an inner derivative appears.

0281 / Calculus

Fundamental theorem of calculus

\[\int_a^b f'(x)\,dx = f(b) - f(a)\]

A definite integral of a derivative equals net change.

0282 / Calculus

Average value of a function

\[f_{avg} = \frac{1}{b-a}\int_a^b f(x)\,dx\]

Average the function over an interval with a definite integral.

0283 / Calculus

Arc length

\[L = \int_a^b \sqrt{1 + \left(f'(x)\right)^2}\,dx\]

Use this to find the length of a smooth curve.

0284 / Calculus

Surface of revolution

\[S = 2\pi\int_a^b f(x)\sqrt{1 + \left(f'(x)\right)^2}\,dx\]

Rotate a curve around the x-axis to get a surface area formula.

0285 / Calculus

Disk method volume

\[V = \pi\int_a^b \left(R(x)\right)^2\,dx\]

Use the disk method when the solid has no hollow center.

0286 / Calculus

Washer method volume

\[V = \pi\int_a^b \left(R(x)^2-r(x)^2\right)\,dx\]

Use the washer method when there is an inner radius.

0287 / Calculus

Shell method volume

\[V = 2\pi\int_a^b x f(x)\,dx\]

Use cylindrical shells for rotation around the y-axis.

0288 / Calculus

Limit of a quotient

\[\lim_{x \to a}\frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}\]

This works when the denominator limit is not zero.

0289 / Calculus

L'Hospital's rule

\[\lim_{x \to a}\frac{f(x)}{g(x)} = \lim_{x \to a}\frac{f'(x)}{g'(x)}\]

Use this on indeterminate forms such as 0/0 when conditions are met.

0290 / Calculus

Maclaurin series for e^x

\[e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}\]

The Maclaurin series expands e^x around zero.

0291 / Calculus

Maclaurin series for sin x

\[\sin x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!}\]

The sine series alternates odd powers.

0292 / Calculus

Maclaurin series for cos x

\[\cos x = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!}\]

The cosine series alternates even powers.

0293 / Calculus

Taylor series

\[f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n\]

Taylor series approximate a smooth function near x = a.

Probability and Statistics

45 formulas

0294 / Probability and Statistics

Factorial

\[n! = n(n-1)(n-2)\cdots 1\]

Factorials count arrangements and appear in permutations and combinations.

0295 / Probability and Statistics

Permutation

\[P(n,r) = \frac{n!}{(n-r)!}\]

Permutations count ordered selections.

0296 / Probability and Statistics

Combination

\[C(n,r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}\]

Combinations count unordered selections.

0297 / Probability and Statistics

Probability of an event

\[P(E) = \frac{\text{favorable outcomes}}{\text{total outcomes}}\]

This is the classical probability formula.

0298 / Probability and Statistics

Complement rule

\[P(E^c) = 1 - P(E)\]

Subtract the event probability from one.

0299 / Probability and Statistics

Addition rule

\[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]

Subtract the overlap to avoid double counting.

0300 / Probability and Statistics

Multiplication rule

\[P(A \cap B) = P(A)P(B|A)\]

Use conditional probability when events are dependent.

0301 / Probability and Statistics

Independent events rule

\[P(A \cap B) = P(A)P(B)\]

Multiply probabilities for independent events.

0302 / Probability and Statistics

Conditional probability

\[P(A|B) = \frac{P(A \cap B)}{P(B)}\]

Conditional probability limits the sample space to B.

0303 / Probability and Statistics

Bayes theorem

\[P(A|B) = \frac{P(B|A)P(A)}{P(B)}\]

Bayes theorem reverses a conditional probability.

0304 / Probability and Statistics

Expected value

\[E(X) = \sum x_i p_i\]

The expected value is the weighted average outcome.

0305 / Probability and Statistics

Variance of a random variable

\[\operatorname{Var}(X) = E[(X-\mu)^2]\]

Variance measures spread around the mean.

0306 / Probability and Statistics

Variance shortcut

\[\operatorname{Var}(X) = E(X^2) - \mu^2\]

Use the expected square minus square of the mean.

0307 / Probability and Statistics

Standard deviation

\[\sigma = \sqrt{\operatorname{Var}(X)}\]

Standard deviation is the square root of variance.

0308 / Probability and Statistics

Sample mean

\[\bar{x} = \frac{\sum x_i}{n}\]

The sample mean averages observed values.

0309 / Probability and Statistics

Sample variance

\[s^2 = \frac{\sum (x_i-\bar{x})^2}{n-1}\]

Use n-1 in the denominator for an unbiased sample estimate.

0310 / Probability and Statistics

Sample standard deviation

\[s = \sqrt{\frac{\sum (x_i-\bar{x})^2}{n-1}}\]

Sample standard deviation is the square root of sample variance.

0311 / Probability and Statistics

Z-score

\[z = \frac{x-\mu}{\sigma}\]

A z-score measures how many standard deviations x is from the mean.

0312 / Probability and Statistics

Standard error of the mean

\[SE = \frac{\sigma}{\sqrt{n}}\]

Standard error shrinks as sample size grows.

0313 / Probability and Statistics

Correlation coefficient

\[r = \frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum (x_i-\bar{x})^2 \sum (y_i-\bar{y})^2}}\]

Correlation measures linear association.

0314 / Probability and Statistics

Covariance

\[\operatorname{Cov}(X,Y) = E[(X-\mu_X)(Y-\mu_Y)]\]

Covariance measures how two variables move together.

0315 / Probability and Statistics

Simple linear regression slope

\[b_1 = r\frac{s_y}{s_x}\]

This gives the slope of the least-squares regression line.

0316 / Probability and Statistics

Simple linear regression intercept

\[b_0 = \bar{y} - b_1\bar{x}\]

The intercept makes the line pass through the sample means.

0317 / Probability and Statistics

Regression prediction

\[\hat{y} = b_0 + b_1x\]

Use the regression equation to estimate y from x.

0318 / Probability and Statistics

Coefficient of determination

\[R^2 = r^2\]

In simple linear regression, R-squared is the squared correlation.

0319 / Probability and Statistics

Binomial probability

\[P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}\]

Use this for repeated independent yes-no trials.

0320 / Probability and Statistics

Binomial mean

\[\mu = np\]

The mean of a binomial distribution is n times p.

0321 / Probability and Statistics

Binomial variance

\[\sigma^2 = np(1-p)\]

The binomial variance depends on both p and 1-p.

0322 / Probability and Statistics

Poisson probability

\[P(X=k) = \frac{e^{-\lambda}\lambda^k}{k!}\]

The Poisson distribution models event counts over time or space.

0323 / Probability and Statistics

Poisson mean

\[\mu = \lambda\]

The mean of a Poisson distribution equals lambda.

0324 / Probability and Statistics

Poisson variance

\[\sigma^2 = \lambda\]

The variance of a Poisson distribution also equals lambda.

0325 / Probability and Statistics

Normal distribution density

\[f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}\]

This is the probability density function of a normal distribution.

0326 / Probability and Statistics

Empirical rule

\[68\%-95\%-99.7\%\]

For a normal distribution, about 68%, 95%, and 99.7% of values fall within one, two, and three standard deviations.

0327 / Probability and Statistics

Confidence interval for a mean

\[\bar{x} \pm z^*\frac{\sigma}{\sqrt{n}}\]

Use this when the population standard deviation is known.

0328 / Probability and Statistics

Confidence interval for a proportion

\[\hat{p} \pm z^*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\]

This interval estimates a population proportion.

0329 / Probability and Statistics

Margin of error for a mean

\[ME = z^*\frac{\sigma}{\sqrt{n}}\]

Margin of error is half the full confidence interval width.

0330 / Probability and Statistics

Margin of error for a proportion

\[ME = z^*\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\]

This margin of error applies to confidence intervals for proportions.

0331 / Probability and Statistics

t-statistic

\[t = \frac{\bar{x}-\mu}{s/\sqrt{n}}\]

Use a t-statistic when the population standard deviation is unknown.

0332 / Probability and Statistics

Chi-square statistic

\[\chi^2 = \sum \frac{(O-E)^2}{E}\]

Compare observed and expected counts with the chi-square test.

0333 / Probability and Statistics

F-statistic

\[F = \frac{s_1^2}{s_2^2}\]

The F-statistic compares two sample variances.

0334 / Probability and Statistics

Coefficient of variation

\[CV = \frac{\sigma}{\mu} \cdot 100\]

The coefficient of variation compares spread relative to the mean.

0335 / Probability and Statistics

Skewness

\[\gamma_1 = \frac{E[(X-\mu)^3]}{\sigma^3}\]

Skewness measures asymmetry in a distribution.

0336 / Probability and Statistics

Kurtosis

\[\gamma_2 = \frac{E[(X-\mu)^4]}{\sigma^4}\]

Kurtosis measures tail weight and peak shape.

0337 / Probability and Statistics

Median position

\[\frac{n+1}{2}\]

For ordered data, this locates the median observation.

0338 / Probability and Statistics

Quartile position

\[Q_k = \frac{k(n+1)}{4}\]

Use quartile positions on ordered data.

Linear Algebra and Vectors

25 formulas

0339 / Linear Algebra and Vectors

Vector magnitude in 2D

\[|\mathbf{v}| = \sqrt{x^2+y^2}\]

Use the Pythagorean theorem on the vector components.

0340 / Linear Algebra and Vectors

Vector magnitude in 3D

\[|\mathbf{v}| = \sqrt{x^2+y^2+z^2}\]

Add the squares of all three components.

0341 / Linear Algebra and Vectors

Unit vector

\[\hat{\mathbf{v}} = \frac{\mathbf{v}}{|\mathbf{v}|}\]

Divide a vector by its magnitude to normalize it.

0342 / Linear Algebra and Vectors

Dot product

\[\mathbf{a}\cdot\mathbf{b} = a_xb_x + a_yb_y + a_zb_z\]

The dot product combines component-wise products.

0343 / Linear Algebra and Vectors

Dot product from angle

\[\mathbf{a}\cdot\mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos\theta\]

Use this form when magnitudes and the included angle are known.

0344 / Linear Algebra and Vectors

Angle between vectors

\[\theta = \cos^{-1}\left(\frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}||\mathbf{b}|}\right)\]

Solve the dot-product formula for the angle.

0345 / Linear Algebra and Vectors

Cross product magnitude

\[|\mathbf{a}\times\mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta\]

The cross product magnitude equals the parallelogram area.

0346 / Linear Algebra and Vectors

Area of a parallelogram from vectors

\[A = |\mathbf{a}\times\mathbf{b}|\]

Use the cross product magnitude.

0347 / Linear Algebra and Vectors

Area of a triangle from vectors

\[A = \frac{1}{2}|\mathbf{a}\times\mathbf{b}|\]

A triangle is half the parallelogram built on the same sides.

0348 / Linear Algebra and Vectors

Vector projection

\[\operatorname{proj}_{\mathbf{b}}\mathbf{a} = \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|^2}\mathbf{b}\]

Project vector a onto vector b.

0349 / Linear Algebra and Vectors

Scalar projection

\[\operatorname{comp}_{\mathbf{b}}\mathbf{a} = \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|}\]

The scalar projection is the signed length on b.

0350 / Linear Algebra and Vectors

2x2 determinant

\[\det\begin{pmatrix}a&b\\c&d\end{pmatrix} = ad-bc\]

Use this for area scaling, invertibility, and Cramer's rule.

0351 / Linear Algebra and Vectors

2x2 matrix inverse

\[A^{-1} = \frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\]

A 2x2 matrix is invertible when its determinant is nonzero.

0352 / Linear Algebra and Vectors

Trace of a matrix

\[\operatorname{tr}(A) = \sum a_{ii}\]

The trace is the sum of diagonal entries.

0353 / Linear Algebra and Vectors

Matrix multiplication entry

\[(AB)_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj}\]

Multiply rows of A by columns of B.

0354 / Linear Algebra and Vectors

Cramer's rule for x

\[x = \frac{\det(A_x)}{\det(A)}\]

Replace the x-column with constants to solve for x.

0355 / Linear Algebra and Vectors

Cramer's rule for y

\[y = \frac{\det(A_y)}{\det(A)}\]

Replace the y-column with constants to solve for y.

0356 / Linear Algebra and Vectors

Eigenvalue equation

\[A\mathbf{v} = \lambda \mathbf{v}\]

An eigenvector keeps its direction under the matrix transformation.

0357 / Linear Algebra and Vectors

Characteristic equation

\[\det(A-\lambda I) = 0\]

Solve this equation to find eigenvalues.

0358 / Linear Algebra and Vectors

Vector equation of a line

\[\mathbf{r} = \mathbf{r}_0 + t\mathbf{v}\]

Use a point vector and a direction vector.

0359 / Linear Algebra and Vectors

Parametric line equations

\[x = x_0 + at,\ y = y_0 + bt,\ z = z_0 + ct\]

Parametric equations describe a line in space.

0360 / Linear Algebra and Vectors

Plane equation from a normal vector

\[a(x-x_0)+b(y-y_0)+c(z-z_0)=0\]

A normal vector defines the orientation of a plane.

0361 / Linear Algebra and Vectors

Distance from a point to a plane

\[d = \frac{|ax_0+by_0+cz_0+d|}{\sqrt{a^2+b^2+c^2}}\]

Divide the plane expression by the normal vector magnitude.

0362 / Linear Algebra and Vectors

Gram-Schmidt projection step

\[\mathbf{u}_2 = \mathbf{v}_2 - \operatorname{proj}_{\mathbf{u}_1}\mathbf{v}_2\]

Subtract the component already covered by the first basis vector.

0363 / Linear Algebra and Vectors

Linear interpolation

\[L(t) = (1-t)a + tb\]

Linear interpolation blends between two endpoints.

Analytic Geometry

15 formulas

0364 / Analytic Geometry

Distance from a point to a line

\[d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2+B^2}}\]

Use the standard-form line coefficients and the point coordinates.

0365 / Analytic Geometry

Intercept form of a line

\[\frac{x}{a} + \frac{y}{b} = 1\]

This line form uses the x-intercept a and y-intercept b.

0366 / Analytic Geometry

Condition for parallel lines

\[m_1 = m_2\]

Two nonvertical lines are parallel when their slopes match.

0367 / Analytic Geometry

Condition for perpendicular lines

\[m_1m_2 = -1\]

Two nonvertical lines are perpendicular when their slopes multiply to negative one.

0368 / Analytic Geometry

Parabola focal length

\[p = \frac{1}{4a}\]

For \(y = ax^2\), the focus is \(\left(0,\frac{1}{4a}\right)\).

0369 / Analytic Geometry

Parabola focus

\[Focus = (h, k+p)\]

For \((x-h)^2 = 4p(y-k)\), the focus lies p units above the vertex.

0370 / Analytic Geometry

Parabola directrix

\[Directrix: y = k-p\]

For a vertical parabola, the directrix lies p units below the vertex.

0371 / Analytic Geometry

Length of latus rectum of a parabola

\[L = 4p\]

The latus rectum spans four focal-length units.

0372 / Analytic Geometry

Hyperbola asymptotes

\[y-k = \pm \frac{b}{a}(x-h)\]

Use these lines to sketch a horizontal hyperbola.

0373 / Analytic Geometry

Ellipse major-axis length

\[Major\ Axis = 2a\]

Double the semi-major axis to get the full major-axis length.

0374 / Analytic Geometry

Ellipse minor-axis length

\[Minor\ Axis = 2b\]

Double the semi-minor axis to get the full minor-axis length.

0375 / Analytic Geometry

Polar to Cartesian x

\[x = r\cos\theta\]

Convert a polar point to its x-coordinate.

0376 / Analytic Geometry

Polar to Cartesian y

\[y = r\sin\theta\]

Convert a polar point to its y-coordinate.

0377 / Analytic Geometry

Cartesian to polar radius

\[r = \sqrt{x^2+y^2}\]

Use the distance from the origin to find the polar radius.

0378 / Analytic Geometry

Cartesian to polar angle

\[\theta = \tan^{-1}\left(\frac{y}{x}\right)\]

Use the quadrant-aware inverse tangent to recover the polar angle.

Discrete Math and Number Theory

15 formulas

0379 / Discrete Math and Number Theory

Sum of the first n integers

\[\sum_{k=1}^{n} k = \frac{n(n+1)}{2}\]

This is the classic triangular-number formula.

0380 / Discrete Math and Number Theory

Sum of the first n squares

\[\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}\]

Use this when adding consecutive square numbers.

0381 / Discrete Math and Number Theory

Sum of the first n cubes

\[\sum_{k=1}^{n} k^3 = \left(\frac{n(n+1)}{2}\right)^2\]

The sum of cubes equals the square of the triangular number.

0382 / Discrete Math and Number Theory

Binomial theorem

\[(x+y)^n = \sum_{k=0}^{n} \binom{n}{k}x^{n-k}y^k\]

Expand any positive-integer binomial power with binomial coefficients.

0383 / Discrete Math and Number Theory

Pascal identity

\[\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}\]

Each Pascal-triangle entry is the sum of the two above it.

0384 / Discrete Math and Number Theory

Permutations with repetition

\[\frac{n!}{n_1!n_2!\cdots n_k!}\]

Use this when some objects are repeated in a permutation count.

0385 / Discrete Math and Number Theory

Multinomial coefficient

\[\binom{n}{n_1,n_2,\ldots,n_k} = \frac{n!}{n_1!n_2!\cdots n_k!}\]

The multinomial coefficient counts grouped arrangements.

0386 / Discrete Math and Number Theory

Greatest common divisor relation

\[\gcd(a,b) = \gcd(b, a \bmod b)\]

The Euclidean algorithm repeatedly replaces the larger number with the remainder.

0387 / Discrete Math and Number Theory

Least common multiple relation

\[\operatorname{lcm}(a,b) = \frac{|ab|}{\gcd(a,b)}\]

LCM and GCD are linked through the product of the two numbers.

0388 / Discrete Math and Number Theory

Modular congruence definition

\[a \equiv b \pmod n \iff n \mid (a-b)\]

Two integers are congruent mod n when their difference is divisible by n.

0389 / Discrete Math and Number Theory

Euler totient for a prime

\[\varphi(p) = p-1\]

A prime number has p-1 positive integers less than p that are coprime to it.

0390 / Discrete Math and Number Theory

Fermat's little theorem

\[a^{p-1} \equiv 1 \pmod p\]

For prime p and a not divisible by p, powers cycle modulo p.

0391 / Discrete Math and Number Theory

Binet formula for Fibonacci numbers

\[F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}\]

This closed form generates Fibonacci numbers from powers of the golden ratio and its conjugate.

0392 / Discrete Math and Number Theory

Catalan number

\[C_n = \frac{1}{n+1}\binom{2n}{n}\]

Catalan numbers count many recursive structures such as balanced parentheses.

0393 / Discrete Math and Number Theory

Inclusion-exclusion for three sets

\[|A\cup B\cup C| = |A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|\]

Inclusion-exclusion corrects for overlaps when counting union sizes.

Additional Calculus Reference

7 formulas

0394 / Additional Calculus Reference

Definite integral power rule

\[\int_a^b x^n\,dx = \frac{b^{n+1} - a^{n+1}}{n+1}\]

Apply the antiderivative to the upper and lower limits when n is not -1.

0395 / Additional Calculus Reference

Integral of (ax+b)^n

\[\int (ax+b)^n\,dx = \frac{(ax+b)^{n+1}}{a(n+1)} + C\]

Use substitution or reverse-chain-rule reasoning for a linear inner expression.

0396 / Additional Calculus Reference

Integral of a negative power

\[\int x^{-n}\,dx = \frac{x^{-n+1}}{-n+1} + C\]

Negative powers still use the power-rule antiderivative when n is not 1.

0397 / Additional Calculus Reference

Integral of a fractional power

\[\int x^{1/n}\,dx = \frac{n}{n+1}x^{\frac{n+1}{n}} + C\]

Fractional powers integrate with the same general power-rule pattern.

0398 / Additional Calculus Reference

Left Riemann sum

\[L_n = \sum_{i=1}^{n} f(x_{i-1})\Delta x\]

A left-endpoint Riemann sum approximates area using the function value at the start of each subinterval.

0399 / Additional Calculus Reference

Midpoint rule

\[M_n = \sum_{i=1}^{n} f\left(\frac{x_{i-1}+x_i}{2}\right)\Delta x\]

The midpoint rule samples each subinterval at its center for a more balanced area estimate.

0400 / Additional Calculus Reference

Trapezoidal rule

\[T_n = \frac{\Delta x}{2}\left[f(x_0) + 2\sum_{i=1}^{n-1} f(x_i) + f(x_n)\right]\]

The trapezoidal rule connects sample points with line segments and sums trapezoid areas.

Mechanics

62 formulas

0401 / Mechanics

Average velocity

\[v_{avg} = \frac{\Delta x}{\Delta t}\]

Average velocity is displacement divided by elapsed time.

0402 / Mechanics

Average acceleration

\[a_{avg} = \frac{\Delta v}{\Delta t}\]

Average acceleration is change in velocity divided by time.

0403 / Mechanics

Velocity from constant acceleration

\[v = u + at\]

Use this for motion with constant acceleration.

0404 / Mechanics

Displacement from constant acceleration

\[s = ut + \frac{1}{2}at^2\]

Initial velocity contributes linearly, acceleration quadratically.

0405 / Mechanics

Velocity-squared equation

\[v^2 = u^2 + 2as\]

This eliminates time from constant-acceleration motion.

0406 / Mechanics

Displacement from average velocity

\[s = \frac{u+v}{2}t\]

Average initial and final velocity when acceleration is constant.

0407 / Mechanics

Acceleration from velocities and time

\[a = \frac{v-u}{t}\]

Rearrange the first kinematics equation.

0408 / Mechanics

Time from kinematics

\[t = \frac{v-u}{a}\]

Solve the velocity equation for time.

0409 / Mechanics

Newton's second law

\[F = ma\]

Net force equals mass times acceleration.

0410 / Mechanics

Acceleration from force

\[a = \frac{F}{m}\]

Divide net force by mass.

0411 / Mechanics

Mass from force

\[m = \frac{F}{a}\]

Solve Newton's second law for mass.

0412 / Mechanics

Weight

\[W = mg\]

Weight is the gravitational force on a mass.

0413 / Mechanics

Momentum

\[p = mv\]

Momentum combines mass and velocity.

0414 / Mechanics

Impulse

\[J = F\Delta t\]

Impulse is force applied over a time interval.

0415 / Mechanics

Impulse-momentum theorem

\[J = \Delta p\]

Impulse equals the change in momentum.

0416 / Mechanics

Work

\[W = Fd\cos\theta\]

Work is the component of force along the displacement times distance.

0417 / Mechanics

Power

\[P = \frac{W}{t}\]

Power is work done per unit time.

0418 / Mechanics

Mechanical power from force and velocity

\[P = Fv\]

When force and velocity are aligned, power equals force times speed.

0419 / Mechanics

Kinetic energy

\[KE = \frac{1}{2}mv^2\]

Kinetic energy depends on mass and the square of speed.

0420 / Mechanics

Gravitational potential energy

\[PE = mgh\]

Near Earth's surface, potential energy depends on height.

0421 / Mechanics

Elastic potential energy

\[U = \frac{1}{2}kx^2\]

Spring energy grows with the square of displacement.

0422 / Mechanics

Hooke's law

\[F = kx\]

Spring force is proportional to displacement from equilibrium.

0423 / Mechanics

Spring constant

\[k = \frac{F}{x}\]

Solve Hooke's law for k.

0424 / Mechanics

Conservation of mechanical energy

\[KE_i + PE_i = KE_f + PE_f\]

In the absence of nonconservative forces, total mechanical energy stays constant.

0425 / Mechanics

Friction force

\[f = \mu N\]

Friction is the coefficient times the normal force.

0426 / Mechanics

Pressure

\[P = \frac{F}{A}\]

Pressure is force distributed over an area.

0427 / Mechanics

Torque

\[\tau = rF\sin\theta\]

Torque measures rotational turning effect.

0428 / Mechanics

Angular displacement

\[\theta = \frac{s}{r}\]

Arc length divided by radius gives angular displacement in radians.

0429 / Mechanics

Angular velocity

\[\omega = \frac{\Delta\theta}{\Delta t}\]

Angular velocity is angle change per unit time.

0430 / Mechanics

Angular acceleration

\[\alpha = \frac{\Delta\omega}{\Delta t}\]

Angular acceleration measures change in angular velocity.

0431 / Mechanics

Tangential velocity

\[v = r\omega\]

Linear speed along a circular path equals radius times angular speed.

0432 / Mechanics

Centripetal acceleration

\[a_c = \frac{v^2}{r}\]

This is the inward acceleration in circular motion.

0433 / Mechanics

Centripetal force

\[F_c = \frac{mv^2}{r}\]

This is the inward force required for circular motion.

0434 / Mechanics

Rotational kinetic energy

\[KE_{rot} = \frac{1}{2}I\omega^2\]

Rotational energy depends on moment of inertia and angular speed.

0435 / Mechanics

Angular momentum

\[L = I\omega\]

Rotational momentum equals moment of inertia times angular speed.

0436 / Mechanics

Angular impulse

\[\tau \Delta t = \Delta L\]

Torque applied over time changes angular momentum.

0437 / Mechanics

Moment of inertia of a point mass

\[I = mr^2\]

Moment of inertia grows with distance from the axis squared.

0438 / Mechanics

Moment of inertia of a solid disk

\[I = \frac{1}{2}mr^2\]

Use this for a uniform solid disk about its center.

0439 / Mechanics

Moment of inertia of a hoop

\[I = mr^2\]

Use this for a thin hoop about its center.

0440 / Mechanics

Moment of inertia of a rod about center

\[I = \frac{1}{12}mL^2\]

Use this for a uniform rod rotated about its midpoint.

0441 / Mechanics

Moment of inertia of a rod about end

\[I = \frac{1}{3}mL^2\]

Use this for a uniform rod rotated about one end.

0442 / Mechanics

Moment of inertia of a solid sphere

\[I = \frac{2}{5}mr^2\]

Use this for a uniform solid sphere.

0443 / Mechanics

Moment of inertia of a spherical shell

\[I = \frac{2}{3}mr^2\]

Use this for a thin hollow sphere.

0444 / Mechanics

Universal gravitation

\[F = G\frac{m_1m_2}{r^2}\]

Any two masses attract with a force proportional to the product of masses.

0445 / Mechanics

Gravitational field strength

\[g = G\frac{M}{r^2}\]

The field strength near a mass depends on mass and distance.

0446 / Mechanics

Orbital velocity

\[v = \sqrt{\frac{GM}{r}}\]

Circular orbital speed balances gravity and centripetal force.

0447 / Mechanics

Escape velocity

\[v_e = \sqrt{\frac{2GM}{r}}\]

Escape velocity is the minimum speed needed to leave a gravitational field.

0448 / Mechanics

Kepler's third law

\[T^2 = \frac{4\pi^2}{GM}r^3\]

Orbital period depends on orbital radius and the central mass.

0449 / Mechanics

Average density of a sphere

\[\rho = \frac{3m}{4\pi r^3}\]

Combine sphere volume with density.

0450 / Mechanics

Projectile range

\[R = \frac{v_0^2\sin(2\theta)}{g}\]

For launch and landing at the same height, horizontal range depends on launch speed and angle.

0451 / Mechanics

Projectile maximum height

\[H = \frac{v_0^2\sin^2\theta}{2g}\]

The vertical component controls maximum height.

0452 / Mechanics

Projectile time of flight

\[T = \frac{2v_0\sin\theta}{g}\]

For a projectile landing at launch height, total time depends on the vertical component.

0453 / Mechanics

Buoyant force

\[F_b = \rho gV\]

Buoyant force equals the weight of displaced fluid.

0454 / Mechanics

Simple harmonic motion displacement

\[x = A\cos(\omega t + \phi)\]

Displacement in simple harmonic motion varies sinusoidally.

0455 / Mechanics

Simple harmonic motion velocity

\[v = -A\omega\sin(\omega t + \phi)\]

Differentiate the displacement function to get velocity.

0456 / Mechanics

Simple harmonic motion acceleration

\[a = -\omega^2 x\]

Acceleration points toward equilibrium and is proportional to displacement.

0457 / Mechanics

Spring-mass angular frequency

\[\omega = \sqrt{\frac{k}{m}}\]

Angular frequency of a spring-mass system depends on stiffness and mass.

0458 / Mechanics

Pendulum period

\[T = 2\pi\sqrt{\frac{L}{g}}\]

For small angles, period depends on pendulum length and gravity.

0459 / Mechanics

Wave speed

\[v = f\lambda\]

Wave speed equals frequency times wavelength.

0460 / Mechanics

Frequency-period relation

\[f = \frac{1}{T}\]

Frequency is the reciprocal of period.

0461 / Mechanics

Angular frequency

\[\omega = 2\pi f\]

Convert cycles per second to radians per second.

0462 / Mechanics

Resonant frequency of a spring

\[f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}\]

This is the frequency form of the spring-mass angular frequency.

Thermodynamics and Fluids

40 formulas

0463 / Thermodynamics and Fluids

Heat transfer

\[Q = mc\Delta T\]

Heat energy depends on mass, specific heat, and temperature change.

0464 / Thermodynamics and Fluids

Specific heat

\[c = \frac{Q}{m\Delta T}\]

Solve the heat formula for specific heat.

0465 / Thermodynamics and Fluids

Latent heat

\[Q = mL\]

Phase-change energy depends on mass and latent heat.

0466 / Thermodynamics and Fluids

First law of thermodynamics

\[\Delta U = Q - W\]

Internal energy changes by heat added minus work done by the system.

0467 / Thermodynamics and Fluids

Ideal gas law

\[PV = nRT\]

This links pressure, volume, amount of gas, and temperature.

0468 / Thermodynamics and Fluids

Pressure from ideal gas law

\[P = \frac{nRT}{V}\]

Solve the ideal gas law for pressure.

0469 / Thermodynamics and Fluids

Volume from ideal gas law

\[V = \frac{nRT}{P}\]

Solve the ideal gas law for volume.

0470 / Thermodynamics and Fluids

Amount of substance from ideal gas law

\[n = \frac{PV}{RT}\]

Solve the ideal gas law for moles.

0471 / Thermodynamics and Fluids

Combined gas law

\[\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}\]

Use this when the amount of gas stays constant.

0472 / Thermodynamics and Fluids

Boyle's law

\[P_1V_1 = P_2V_2\]

At constant temperature, pressure and volume are inversely related.

0473 / Thermodynamics and Fluids

Charles's law

\[\frac{V_1}{T_1} = \frac{V_2}{T_2}\]

At constant pressure, volume is proportional to absolute temperature.

0474 / Thermodynamics and Fluids

Gay-Lussac's law

\[\frac{P_1}{T_1} = \frac{P_2}{T_2}\]

At constant volume, pressure is proportional to absolute temperature.

0475 / Thermodynamics and Fluids

Density of an ideal gas

\[\rho = \frac{PM}{RT}\]

Use molar mass M to convert from amount to mass density.

0476 / Thermodynamics and Fluids

Thermal efficiency

\[\eta = \frac{W_{out}}{Q_{in}}\]

Heat-engine efficiency compares useful work to heat input.

0477 / Thermodynamics and Fluids

Carnot efficiency

\[\eta_C = 1 - \frac{T_c}{T_h}\]

Maximum possible heat-engine efficiency depends only on reservoir temperatures.

0478 / Thermodynamics and Fluids

Coefficient of performance for a refrigerator

\[COP_R = \frac{Q_c}{W}\]

Refrigerator performance compares heat removed to work input.

0479 / Thermodynamics and Fluids

Coefficient of performance for a heat pump

\[COP_{HP} = \frac{Q_h}{W}\]

Heat-pump performance compares heat delivered to work input.

0480 / Thermodynamics and Fluids

Hydrostatic pressure

\[P = P_0 + \rho gh\]

Fluid pressure increases with depth.

0481 / Thermodynamics and Fluids

Gauge pressure

\[P_g = \rho gh\]

Gauge pressure measures excess pressure above atmospheric pressure.

0482 / Thermodynamics and Fluids

Continuity equation

\[A_1v_1 = A_2v_2\]

For incompressible steady flow, volume flow rate stays constant.

0483 / Thermodynamics and Fluids

Volume flow rate

\[Q = Av\]

Flow rate equals area times fluid speed.

0484 / Thermodynamics and Fluids

Bernoulli equation

\[P + \frac{1}{2}\rho v^2 + \rho gh = constant\]

Fluid pressure, kinetic energy density, and potential energy density trade off along a streamline.

0485 / Thermodynamics and Fluids

Pascal's principle

\[\frac{F_1}{A_1} = \frac{F_2}{A_2}\]

Pressure applied to an enclosed fluid is transmitted equally.

0486 / Thermodynamics and Fluids

Reynolds number

\[Re = \frac{\rho vD}{\mu}\]

Reynolds number estimates whether flow is laminar or turbulent.

0487 / Thermodynamics and Fluids

Dynamic viscosity

\[\tau = \mu \frac{du}{dy}\]

Shear stress in a fluid is proportional to the velocity gradient.

0488 / Thermodynamics and Fluids

Poiseuille's law

\[Q = \frac{\pi r^4\Delta P}{8\mu L}\]

This gives laminar flow through a cylindrical pipe.

0489 / Thermodynamics and Fluids

Stokes drag

\[F_d = 6\pi\mu rv\]

Use this for a small sphere moving slowly through a viscous fluid.

0490 / Thermodynamics and Fluids

Thermal expansion of length

\[\Delta L = \alpha L_0\Delta T\]

Linear expansion depends on original length, temperature change, and linear expansion coefficient.

0491 / Thermodynamics and Fluids

Thermal expansion of area

\[\Delta A = 2\alpha A_0\Delta T\]

Area expansion is approximately twice the linear coefficient.

0492 / Thermodynamics and Fluids

Thermal expansion of volume

\[\Delta V = \beta V_0\Delta T\]

Volume expansion depends on the volumetric coefficient.

0493 / Thermodynamics and Fluids

Conduction rate

\[\frac{Q}{t} = kA\frac{\Delta T}{L}\]

Heat conduction increases with conductivity, area, and temperature difference.

0494 / Thermodynamics and Fluids

Stefan-Boltzmann law

\[P = \sigma A T^4\]

A blackbody radiates power proportional to the fourth power of temperature.

0495 / Thermodynamics and Fluids

Wien's displacement law

\[\lambda_{max} T = b\]

Hotter blackbodies peak at shorter wavelengths.

0496 / Thermodynamics and Fluids

Specific gravity

\[SG = \frac{\rho_{substance}}{\rho_{water}}\]

Specific gravity compares density to water.

0497 / Thermodynamics and Fluids

Bulk modulus

\[B = -\frac{\Delta P}{\Delta V / V}\]

Bulk modulus measures resistance to compression.

0498 / Thermodynamics and Fluids

Young's modulus

\[E = \frac{Stress}{Strain}\]

Young's modulus measures stiffness in tension or compression.

0499 / Thermodynamics and Fluids

Stress

\[\sigma = \frac{F}{A}\]

Stress is force per unit area.

0500 / Thermodynamics and Fluids

Strain

\[\epsilon = \frac{\Delta L}{L}\]

Strain is relative change in length.

0501 / Thermodynamics and Fluids

Shear modulus

\[G = \frac{Shear\ Stress}{Shear\ Strain}\]

Shear modulus measures resistance to shape change.

0502 / Thermodynamics and Fluids

Spring energy density

\[u = \frac{1}{2}\sigma\epsilon\]

Energy stored elastically per unit volume equals half stress times strain.

Waves and Optics

27 formulas

0503 / Waves and Optics

Wave speed relation

\[v = f\lambda\]

Wave speed equals frequency times wavelength.

0504 / Waves and Optics

Period from frequency

\[T = \frac{1}{f}\]

Period is the reciprocal of frequency.

0505 / Waves and Optics

String wave speed

\[v = \sqrt{\frac{T}{\mu}}\]

Wave speed on a string depends on tension and linear density.

0506 / Waves and Optics

Intensity

\[I = \frac{P}{A}\]

Intensity is power spread over an area.

0507 / Waves and Optics

Inverse-square intensity

\[I = \frac{P}{4\pi r^2}\]

Spherical spreading reduces intensity with the square of distance.

0508 / Waves and Optics

Snell's law

\[n_1\sin\theta_1 = n_2\sin\theta_2\]

Light bends when it enters a medium with a different refractive index.

0509 / Waves and Optics

Index of refraction

\[n = \frac{c}{v}\]

A material's index compares vacuum light speed to light speed in the material.

0510 / Waves and Optics

Mirror equation

\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]

Mirror focal length, object distance, and image distance are related.

0511 / Waves and Optics

Lens equation

\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]

Thin lenses follow the same distance relation as mirrors.

0512 / Waves and Optics

Magnification

\[m = -\frac{d_i}{d_o} = \frac{h_i}{h_o}\]

Magnification compares image size to object size.

0513 / Waves and Optics

Power of a lens

\[P = \frac{1}{f}\]

Lens power is the reciprocal of focal length in meters.

0514 / Waves and Optics

Critical angle

\[\sin\theta_c = \frac{n_2}{n_1}\]

This applies when light moves from a denser to a rarer medium.

0515 / Waves and Optics

Double-slit fringe spacing

\[\Delta y = \frac{\lambda L}{d}\]

Fringe spacing depends on wavelength, screen distance, and slit separation.

0516 / Waves and Optics

Single-slit diffraction minimum

\[a\sin\theta = m\lambda\]

Minima occur where path differences cancel.

0517 / Waves and Optics

Bragg's law

\[2d\sin\theta = n\lambda\]

Bragg's law describes constructive interference in crystals.

0518 / Waves and Optics

Doppler effect for sound

\[f' = f\frac{v \pm v_o}{v \mp v_s}\]

Observed frequency shifts when the source or observer moves.

0519 / Waves and Optics

Beat frequency

\[f_b = |f_1-f_2|\]

Close frequencies interfere to produce beats.

0520 / Waves and Optics

Photon energy

\[E = hf\]

A photon's energy is Planck's constant times frequency.

0521 / Waves and Optics

Photon momentum

\[p = \frac{h}{\lambda}\]

A photon's momentum is inversely proportional to wavelength.

0522 / Waves and Optics

de Broglie wavelength

\[\lambda = \frac{h}{p}\]

Matter waves have wavelength inversely related to momentum.

0523 / Waves and Optics

Photoelectric equation

\[K_{max} = hf - \phi\]

Maximum photoelectron kinetic energy equals photon energy minus work function.

0524 / Waves and Optics

Malus's law

\[I = I_0\cos^2\theta\]

Intensity through a polarizer depends on angle relative to the initial polarization.

0525 / Waves and Optics

Young's modulus from wave speed

\[v = \sqrt{\frac{E}{\rho}}\]

Longitudinal wave speed in a solid depends on stiffness and density.

0526 / Waves and Optics

Resonant frequency of an open pipe

\[f_n = \frac{nv}{2L}\]

Open pipes support harmonics at integer multiples.

0527 / Waves and Optics

Resonant frequency of a closed pipe

\[f_n = \frac{nv}{4L}\]

Closed pipes support odd harmonics.

0528 / Waves and Optics

Standing-wave wavelength on a string

\[\lambda_n = \frac{2L}{n}\]

Allowed wavelengths on a string are set by the string length.

0529 / Waves and Optics

Blackbody photon peak energy estimate

\[E_{peak} \approx 2.82kT\]

A thermal spectrum peaks near this energy in a Planck distribution.

Electricity and Magnetism

57 formulas

0530 / Electricity and Magnetism

Charge

\[Q = It\]

Electric charge transferred equals current times time.

0531 / Electricity and Magnetism

Current

\[I = \frac{Q}{t}\]

Current is charge flow per unit time.

0532 / Electricity and Magnetism

Ohm's law

\[V = IR\]

Voltage equals current times resistance.

0533 / Electricity and Magnetism

Current from Ohm's law

\[I = \frac{V}{R}\]

Solve Ohm's law for current.

0534 / Electricity and Magnetism

Resistance from Ohm's law

\[R = \frac{V}{I}\]

Solve Ohm's law for resistance.

0535 / Electricity and Magnetism

Electric power

\[P = VI\]

Electrical power equals voltage times current.

0536 / Electricity and Magnetism

Power from current and resistance

\[P = I^2R\]

Substitute Ohm's law into the power formula.

0537 / Electricity and Magnetism

Power from voltage and resistance

\[P = \frac{V^2}{R}\]

Substitute current from Ohm's law into the power formula.

0538 / Electricity and Magnetism

Electrical energy

\[E = Pt\]

Energy equals power multiplied by time.

0539 / Electricity and Magnetism

Resistance of a wire

\[R = \rho\frac{L}{A}\]

Resistance depends on resistivity, length, and cross-sectional area.

0540 / Electricity and Magnetism

Conductivity-resistivity relation

\[\sigma = \frac{1}{\rho}\]

Conductivity is the reciprocal of resistivity.

0541 / Electricity and Magnetism

Series resistors

\[R_{eq} = R_1 + R_2 + \cdots\]

Resistances in series add directly.

0542 / Electricity and Magnetism

Parallel resistors

\[\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots\]

Parallel paths combine by reciprocal sums.

0543 / Electricity and Magnetism

Capacitance

\[C = \frac{Q}{V}\]

Capacitance is charge stored per unit voltage.

0544 / Electricity and Magnetism

Parallel-plate capacitance

\[C = \epsilon\frac{A}{d}\]

Capacitance grows with plate area and decreases with separation.

0545 / Electricity and Magnetism

Energy stored in a capacitor

\[U = \frac{1}{2}CV^2\]

Capacitors store electric potential energy.

0546 / Electricity and Magnetism

Capacitors in series

\[\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots\]

Series capacitors combine through reciprocal sums.

0547 / Electricity and Magnetism

Capacitors in parallel

\[C_{eq} = C_1 + C_2 + \cdots\]

Parallel capacitors add directly.

0548 / Electricity and Magnetism

Electric field

\[E = \frac{F}{q}\]

Electric field is force per unit charge.

0549 / Electricity and Magnetism

Field from point charge

\[E = k\frac{Q}{r^2}\]

A point charge creates a radial inverse-square field.

0550 / Electricity and Magnetism

Coulomb's law

\[F = k\frac{|q_1q_2|}{r^2}\]

Electric force between point charges follows an inverse-square law.

0551 / Electricity and Magnetism

Electric potential

\[V = \frac{U}{q}\]

Electric potential is potential energy per unit charge.

0552 / Electricity and Magnetism

Point-charge potential

\[V = k\frac{Q}{r}\]

A point charge's electric potential decreases with distance.

0553 / Electricity and Magnetism

Potential difference work relation

\[W = q\Delta V\]

Work done by an electric field equals charge times potential difference.

0554 / Electricity and Magnetism

Gauss's law

\[\Phi_E = \frac{Q_{enc}}{\epsilon_0}\]

Electric flux through a closed surface depends on enclosed charge.

0555 / Electricity and Magnetism

Electric flux

\[\Phi_E = EA\cos\theta\]

Electric flux depends on field strength, area, and orientation.

0556 / Electricity and Magnetism

Force on a current-carrying wire

\[F = BIL\sin\theta\]

A magnetic field exerts force on current in a wire.

0557 / Electricity and Magnetism

Magnetic force on a moving charge

\[F = qvB\sin\theta\]

A magnetic field deflects a moving charge.

0558 / Electricity and Magnetism

Magnetic flux

\[\Phi_B = BA\cos\theta\]

Magnetic flux depends on field strength, area, and orientation.

0559 / Electricity and Magnetism

Faraday's law

\[\mathcal{E} = -N\frac{d\Phi_B}{dt}\]

Changing magnetic flux induces an emf.

0560 / Electricity and Magnetism

Lenz's law sign

\[\mathcal{E} = -N\frac{d\Phi_B}{dt}\]

The negative sign shows the induced emf opposes the change causing it.

0561 / Electricity and Magnetism

Inductance

\[\mathcal{E} = -L\frac{dI}{dt}\]

An inductor resists changes in current.

0562 / Electricity and Magnetism

Energy in an inductor

\[U = \frac{1}{2}LI^2\]

Inductors store magnetic energy.

0563 / Electricity and Magnetism

Magnetic field around a long straight wire

\[B = \frac{\mu_0 I}{2\pi r}\]

Field strength falls off inversely with distance from the wire.

0564 / Electricity and Magnetism

Magnetic field inside a solenoid

\[B = \mu_0 nI\]

An ideal solenoid creates a nearly uniform internal field.

0565 / Electricity and Magnetism

Magnetic force between parallel wires

\[\frac{F}{L} = \frac{\mu_0 I_1 I_2}{2\pi r}\]

Parallel currents attract or repel depending on direction.

0566 / Electricity and Magnetism

Cyclotron frequency

\[f = \frac{qB}{2\pi m}\]

Charged particles orbit in a magnetic field at the cyclotron frequency.

0567 / Electricity and Magnetism

Radius of circular motion for a charge

\[r = \frac{mv}{qB}\]

Balance magnetic force with centripetal force.

0568 / Electricity and Magnetism

RC time constant

\[\tau = RC\]

The time constant controls capacitor charging and discharging speed.

0569 / Electricity and Magnetism

Capacitor charging

\[q(t) = CV\left(1-e^{-t/RC}\right)\]

Charge rises exponentially toward its final value.

0570 / Electricity and Magnetism

Capacitor discharging

\[q(t) = q_0e^{-t/RC}\]

Charge decays exponentially as the capacitor discharges.

0571 / Electricity and Magnetism

RL time constant

\[\tau = \frac{L}{R}\]

The time constant controls current change in an RL circuit.

0572 / Electricity and Magnetism

Current growth in an RL circuit

\[I(t) = \frac{V}{R}\left(1-e^{-tR/L}\right)\]

Current rises exponentially toward its steady-state value.

0573 / Electricity and Magnetism

Current decay in an RL circuit

\[I(t) = I_0e^{-tR/L}\]

Current decays exponentially when the source is removed.

0574 / Electricity and Magnetism

Transformer equation

\[\frac{V_s}{V_p} = \frac{N_s}{N_p}\]

Voltage ratio equals turns ratio in an ideal transformer.

0575 / Electricity and Magnetism

Current ratio in an ideal transformer

\[\frac{I_s}{I_p} = \frac{N_p}{N_s}\]

Current scales inversely with turns ratio in an ideal transformer.

0576 / Electricity and Magnetism

AC rms voltage

\[V_{rms} = \frac{V_0}{\sqrt{2}}\]

Rms voltage of a sinusoidal source equals peak voltage over root two.

0577 / Electricity and Magnetism

AC rms current

\[I_{rms} = \frac{I_0}{\sqrt{2}}\]

Rms current of a sinusoidal source equals peak current over root two.

0578 / Electricity and Magnetism

Average AC power

\[P_{avg} = V_{rms}I_{rms}\cos\phi\]

Power factor accounts for phase difference.

0579 / Electricity and Magnetism

Impedance of a resistor

\[Z_R = R\]

A resistor's impedance equals its resistance.

0580 / Electricity and Magnetism

Inductive reactance

\[X_L = \omega L\]

Reactance of an inductor increases with frequency.

0581 / Electricity and Magnetism

Capacitive reactance

\[X_C = \frac{1}{\omega C}\]

Reactance of a capacitor decreases with frequency.

0582 / Electricity and Magnetism

Series RLC impedance

\[Z = \sqrt{R^2 + (X_L - X_C)^2}\]

Combine resistance and net reactance to get impedance.

0583 / Electricity and Magnetism

Resonant frequency of an RLC circuit

\[f_0 = \frac{1}{2\pi\sqrt{LC}}\]

At resonance, inductive and capacitive reactance cancel.

0584 / Electricity and Magnetism

Lorentz force

\[\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B})\]

A charged particle feels combined electric and magnetic forces.

0585 / Electricity and Magnetism

Drift velocity

\[I = nqAv_d\]

Current in a conductor depends on charge density, area, and drift speed.

0586 / Electricity and Magnetism

Hall voltage

\[V_H = \frac{BIl}{nqt}\]

The Hall effect creates a transverse voltage in a current-carrying conductor within a magnetic field.

Chemistry

51 formulas

0587 / Chemistry

Moles from mass

\[n = \frac{m}{M}\]

Amount of substance equals mass divided by molar mass.

0588 / Chemistry

Mass from moles

\[m = nM\]

Mass equals moles times molar mass.

0589 / Chemistry

Molarity

\[M = \frac{n}{V}\]

Molarity measures moles of solute per liter of solution.

0590 / Chemistry

Molality

\[m = \frac{n_{solute}}{kg_{solvent}}\]

Molality uses kilograms of solvent instead of liters of solution.

0591 / Chemistry

Dilution equation

\[M_1V_1 = M_2V_2\]

Dilution keeps the moles of solute constant.

0592 / Chemistry

Percent yield

\[\%\ Yield = \frac{Actual}{Theoretical} \cdot 100\]

Compare actual product to theoretical maximum.

0593 / Chemistry

Theoretical yield

\[Theoretical\ Yield = Limiting\ Reagent \times Stoichiometric\ Factor\]

Use the balanced equation to convert from limiting reagent to product.

0594 / Chemistry

Empirical formula ratio

\[Ratio = \frac{moles\ of\ element}{smallest\ moles}\]

Divide each mole amount by the smallest to get empirical subscripts.

0595 / Chemistry

Molecular formula factor

\[Factor = \frac{Molar\ Mass}{Empirical\ Formula\ Mass}\]

Multiply empirical subscripts by the factor to get the molecular formula.

0596 / Chemistry

Gas density

\[\rho = \frac{PM}{RT}\]

Gas density comes from the ideal gas law and molar mass.

0597 / Chemistry

Partial pressure

\[P_i = x_i P_{total}\]

A gas's partial pressure equals its mole fraction times the total pressure.

0598 / Chemistry

Mole fraction

\[x_i = \frac{n_i}{n_{total}}\]

Mole fraction expresses a component's share of total moles.

0599 / Chemistry

Raoult's law

\[P_i = x_i P_i^\circ\]

A solvent's vapor pressure decreases in proportion to its mole fraction.

0600 / Chemistry

Boiling-point elevation

\[\Delta T_b = iK_bm\]

Boiling point rises with molality, the van't Hoff factor, and the ebullioscopic constant.

0601 / Chemistry

Freezing-point depression

\[\Delta T_f = iK_fm\]

Freezing point falls with molality, the van't Hoff factor, and the cryoscopic constant.

0602 / Chemistry

Osmotic pressure

\[\Pi = MRT\]

Osmotic pressure depends on solution molarity and absolute temperature.

0603 / Chemistry

pH

\[pH = -\log[H^+]\]

pH measures hydrogen ion concentration on a logarithmic scale.

0604 / Chemistry

pOH

\[pOH = -\log[OH^-]\]

pOH measures hydroxide concentration on a logarithmic scale.

0605 / Chemistry

Water ion-product relation

\[pH + pOH = 14\]

At 25 C, aqueous solutions satisfy this relation.

0606 / Chemistry

Acid dissociation constant

\[K_a = \frac{[H^+][A^-]}{[HA]}\]

Ka measures acid strength at equilibrium.

0607 / Chemistry

Base dissociation constant

\[K_b = \frac{[BH^+][OH^-]}{[B]}\]

Kb measures base strength at equilibrium.

0608 / Chemistry

Henderson-Hasselbalch equation

\[pH = pK_a + \log\frac{[A^-]}{[HA]}\]

Use this for buffer solutions made of a weak acid and its conjugate base.

0609 / Chemistry

Equilibrium constant

\[K = \frac{[Products]^{coeff}}{[Reactants]^{coeff}}\]

Raise each concentration to its stoichiometric coefficient.

0610 / Chemistry

Reaction quotient

\[Q = \frac{[Products]^{coeff}}{[Reactants]^{coeff}}\]

Q has the same form as K but uses current concentrations.

0611 / Chemistry

Gibbs free energy

\[\Delta G = \Delta H - T\Delta S\]

Free energy predicts spontaneity at constant temperature and pressure.

0612 / Chemistry

Free energy from equilibrium constant

\[\Delta G^\circ = -RT\ln K\]

Large K values correspond to negative standard free energy changes.

0613 / Chemistry

Enthalpy from bond energies

\[\Delta H \approx \sum E_{broken} - \sum E_{formed}\]

Estimate reaction enthalpy from bonds broken minus bonds formed.

0614 / Chemistry

Calorimetry

\[q_{lost} = -q_{gained}\]

Heat lost by one part of a system equals heat gained by another.

0615 / Chemistry

Nernst equation

\[E = E^\circ - \frac{RT}{nF}\ln Q\]

Electrode potential shifts with concentration and reaction quotient.

0616 / Chemistry

Faraday's law of electrolysis

\[m = \frac{Q M}{nF}\]

Electrolyzed mass depends on charge passed, molar mass, electron count, and Faraday's constant.

0617 / Chemistry

Cell potential

\[E^\circ_{cell} = E^\circ_{cathode} - E^\circ_{anode}\]

Standard cell voltage equals reduction potential difference.

0618 / Chemistry

Beer-Lambert law

\[A = \epsilon \ell c\]

Absorbance is proportional to molar absorptivity, path length, and concentration.

0619 / Chemistry

Reaction rate

\[Rate = -\frac{1}{a}\frac{d[A]}{dt} = \frac{1}{b}\frac{d[B]}{dt}\]

Stoichiometric coefficients link concentration changes to the reaction rate.

0620 / Chemistry

Rate law

\[Rate = k[A]^m[B]^n\]

Concentration exponents come from experiment, not directly from coefficients.

0621 / Chemistry

Arrhenius equation

\[k = Ae^{-E_a/(RT)}\]

Reaction rate constants increase with temperature.

0622 / Chemistry

Integrated first-order rate law

\[\ln\frac{[A]_t}{[A]_0} = -kt\]

A straight line of ln[A] versus time indicates first-order kinetics.

0623 / Chemistry

Integrated second-order rate law

\[\frac{1}{[A]_t} = \frac{1}{[A]_0} + kt\]

A straight line of 1/[A] versus time indicates second-order kinetics.

0624 / Chemistry

First-order half-life

\[t_{1/2} = \frac{\ln 2}{k}\]

First-order half-life is constant and independent of concentration.

0625 / Chemistry

Second-order half-life

\[t_{1/2} = \frac{1}{k[A]_0}\]

Second-order half-life depends on the initial concentration.

0626 / Chemistry

Clausius-Clapeyron equation

\[\ln\frac{P_2}{P_1} = -\frac{\Delta H_{vap}}{R}\left(\frac{1}{T_2}-\frac{1}{T_1}\right)\]

Relate vapor pressures at two temperatures.

0627 / Chemistry

van't Hoff equation

\[\ln\frac{K_2}{K_1} = -\frac{\Delta H^\circ}{R}\left(\frac{1}{T_2}-\frac{1}{T_1}\right)\]

Relate equilibrium constants at two temperatures.

0628 / Chemistry

Atomic mass average

\[Atomic\ Mass = \sum (Fractional\ Abundance \times Isotopic\ Mass)\]

Average isotopic masses by their natural abundances.

0629 / Chemistry

Percent composition

\[\%\ Element = \frac{Mass\ of\ Element}{Molar\ Mass\ of\ Compound} \cdot 100\]

Percent composition shows how much each element contributes by mass.

0630 / Chemistry

Formal charge

\[FC = Valence - Nonbonding - \frac{Bonding}{2}\]

Formal charge helps compare Lewis structures.

0631 / Chemistry

Bond order

\[Bond\ Order = \frac{Bonding\ Electrons - Antibonding\ Electrons}{2}\]

Bond order indicates bond strength in molecular orbital theory.

0632 / Chemistry

Graham's law

\[\frac{r_1}{r_2} = \sqrt{\frac{M_2}{M_1}}\]

Lighter gases diffuse or effuse faster than heavier gases.

0633 / Chemistry

van der Waals equation

\[\left(P + a\frac{n^2}{V^2}\right)(V-nb) = nRT\]

Correct the ideal gas law for intermolecular attraction and finite molecular volume.

0634 / Chemistry

Buffer capacity approximation

\[\beta = 2.303C\frac{K_a[H^+]}{(K_a+[H^+])^2}\]

Buffer capacity measures resistance to pH change.

0635 / Chemistry

Solubility product

\[K_{sp} = [M^{m+}]^a[X^{n-}]^b\]

Ksp uses the dissolved ion concentrations raised to stoichiometric powers.

0636 / Chemistry

Nuclear decay

\[N = N_0e^{-\lambda t}\]

Radioactive nuclei decay exponentially over time.

0637 / Chemistry

Nuclear half-life

\[t_{1/2} = \frac{\ln 2}{\lambda}\]

Half-life depends on the decay constant.

Biology, Medicine, and Earth Science

20 formulas

0638 / Biology, Medicine, and Earth Science

Exponential population growth

\[N_t = N_0e^{rt}\]

Population size grows exponentially when the per-capita growth rate stays constant.

0639 / Biology, Medicine, and Earth Science

Logistic growth

\[N_t = \frac{K}{1 + Ae^{-rt}}\]

Logistic growth slows as population approaches carrying capacity.

0640 / Biology, Medicine, and Earth Science

Hardy-Weinberg relation

\[p^2 + 2pq + q^2 = 1\]

Allele frequencies in an ideal population determine genotype frequencies.

0641 / Biology, Medicine, and Earth Science

Body mass index

\[BMI = \frac{m}{h^2}\]

BMI compares body mass to height squared.

0642 / Biology, Medicine, and Earth Science

Body surface area (Mosteller)

\[BSA = \sqrt{\frac{Height\times Weight}{3600}}\]

Body surface area is commonly estimated from height and weight.

0643 / Biology, Medicine, and Earth Science

Mean arterial pressure

\[MAP = \frac{SBP + 2DBP}{3}\]

Mean arterial pressure weights diastolic time more heavily than systolic time.

0644 / Biology, Medicine, and Earth Science

Cardiac output

\[CO = HR \times SV\]

Cardiac output equals heart rate times stroke volume.

0645 / Biology, Medicine, and Earth Science

Alveolar ventilation

\[V_A = (V_T - V_D)f\]

Subtract dead-space volume from tidal volume, then multiply by breathing rate.

0646 / Biology, Medicine, and Earth Science

Glomerular filtration pressure

\[NFP = P_{GC} - P_{BS} - \pi_{GC}\]

Net filtration pressure drives kidney filtration.

0647 / Biology, Medicine, and Earth Science

Fick principle for oxygen consumption

\[\dot{V}_{O_2} = CO(C_aO_2 - C_vO_2)\]

Oxygen use equals cardiac output times arterial-venous oxygen difference.

0648 / Biology, Medicine, and Earth Science

Seismic wave speed

\[v = \frac{Distance}{Time}\]

Seismic wave speed comes from travel distance divided by travel time.

0649 / Biology, Medicine, and Earth Science

Earth surface gravity

\[g = \frac{GM_E}{R_E^2}\]

Gravity at Earth's surface depends on Earth mass and radius.

0650 / Biology, Medicine, and Earth Science

Escape velocity from a planet

\[v_e = \sqrt{\frac{2GM}{R}}\]

Escape speed rises with planetary mass and falls with radius.

0651 / Biology, Medicine, and Earth Science

Orbital period

\[T = 2\pi\sqrt{\frac{a^3}{GM}}\]

Orbital period depends on semi-major axis and central mass.

0652 / Biology, Medicine, and Earth Science

Luminosity from radius and temperature

\[L = 4\pi R^2\sigma T^4\]

A star's luminosity depends on emitting area and surface temperature.

0653 / Biology, Medicine, and Earth Science

Distance modulus

\[m - M = 5\log_{10}(d) - 5\]

Astronomers compare apparent and absolute magnitude to estimate distance in parsecs.

0654 / Biology, Medicine, and Earth Science

Redshift

\[z = \frac{\lambda_{obs}-\lambda_{emit}}{\lambda_{emit}}\]

Redshift measures the fractional increase in wavelength.

0655 / Biology, Medicine, and Earth Science

Hubble's law

\[v = H_0 d\]

Galaxies recede faster as their distance increases.

0656 / Biology, Medicine, and Earth Science

Magnification from a microscope

\[M = M_{objective}M_{eyepiece}\]

Total magnification is the product of objective and eyepiece magnifications.

0657 / Biology, Medicine, and Earth Science

Sound level

\[\beta = 10\log_{10}\left(\frac{I}{I_0}\right)\]

Sound level in decibels compares intensity to a reference intensity.

Modern Physics and Relativity

20 formulas

0658 / Modern Physics and Relativity

Mass-energy equivalence

\[E = mc^2\]

Mass and energy are interchangeable according to relativity.

0659 / Modern Physics and Relativity

Relativistic momentum

\[p = \gamma mv\]

Relativistic momentum grows faster than classical momentum at high speeds.

0660 / Modern Physics and Relativity

Lorentz factor

\[\gamma = \frac{1}{\sqrt{1-v^2/c^2}}\]

The Lorentz factor appears throughout special relativity.

0661 / Modern Physics and Relativity

Time dilation

\[\Delta t = \gamma \Delta t_0\]

Moving clocks run slow relative to stationary observers.

0662 / Modern Physics and Relativity

Length contraction

\[L = \frac{L_0}{\gamma}\]

Lengths parallel to motion contract for fast-moving objects.

0663 / Modern Physics and Relativity

Relativistic total energy

\[E^2 = (pc)^2 + (mc^2)^2\]

Energy, momentum, and mass are linked by this invariant relation.

0664 / Modern Physics and Relativity

Photon energy-frequency relation

\[E = hf\]

Photon energy rises linearly with frequency.

0665 / Modern Physics and Relativity

Photon energy-wavelength relation

\[E = \frac{hc}{\lambda}\]

Photon energy is inversely proportional to wavelength.

0666 / Modern Physics and Relativity

Compton wavelength shift

\[\Delta\lambda = \frac{h}{m_ec}(1-\cos\theta)\]

Scattered photons shift wavelength depending on the scattering angle.

0667 / Modern Physics and Relativity

Heisenberg uncertainty principle

\[\Delta x\Delta p \geq \frac{\hbar}{2}\]

Position and momentum cannot both be known exactly.

0668 / Modern Physics and Relativity

Energy-time uncertainty relation

\[\Delta E\Delta t \geq \frac{\hbar}{2}\]

Short-lived states can have uncertain energies.

0669 / Modern Physics and Relativity

Bohr radius

\[a_0 = \frac{4\pi\epsilon_0\hbar^2}{m_ee^2}\]

The Bohr radius sets the characteristic size of hydrogen's ground-state orbit.

0670 / Modern Physics and Relativity

Bohr energy levels

\[E_n = -\frac{13.6\ \text{eV}}{n^2}\]

Hydrogen energy levels get closer together as n increases.

0671 / Modern Physics and Relativity

Rydberg formula

\[\frac{1}{\lambda} = R_H\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)\]

Use the Rydberg formula for spectral lines in hydrogen.

0672 / Modern Physics and Relativity

Nuclear binding energy

\[BE = \Delta m c^2\]

Mass defect converts to binding energy through mass-energy equivalence.

0673 / Modern Physics and Relativity

Activity of a radioactive sample

\[A = \lambda N\]

Activity counts decays per unit time.

0674 / Modern Physics and Relativity

Radioactive decay law

\[N = N_0e^{-\lambda t}\]

Undecayed nuclei decrease exponentially over time.

0675 / Modern Physics and Relativity

Mean lifetime

\[\tau = \frac{1}{\lambda}\]

Mean lifetime is the reciprocal of the decay constant.

0676 / Modern Physics and Relativity

Blackbody Planck relation

\[E_{quantum} = hf\]

Energy is exchanged in discrete quanta.

0677 / Modern Physics and Relativity

Bragg diffraction condition

\[2d\sin\theta = n\lambda\]

Crystal layers produce constructive interference at specific angles.

Circuit and Electromagnetic Reference

25 formulas

0678 / Circuit and Electromagnetic Reference

Kirchhoff's current law

\[\sum I_{in} = \sum I_{out}\]

The total current entering a node equals the total current leaving it.

0679 / Circuit and Electromagnetic Reference

Kirchhoff's voltage law

\[\sum V = 0\]

The directed sum of potential changes around a closed loop is zero.

0680 / Circuit and Electromagnetic Reference

Capacitor current relation

\[I = C\frac{dV}{dt}\]

Capacitor current depends on how quickly voltage changes.

0681 / Circuit and Electromagnetic Reference

Inductor voltage relation

\[V = L\frac{dI}{dt}\]

Inductor voltage depends on how quickly current changes.

0682 / Circuit and Electromagnetic Reference

Electric-field energy density

\[u_E = \frac{1}{2}\epsilon E^2\]

Electric fields store energy in space.

0683 / Circuit and Electromagnetic Reference

Magnetic-field energy density

\[u_B = \frac{B^2}{2\mu}\]

Magnetic fields store energy in space.

0684 / Circuit and Electromagnetic Reference

Motional emf

\[\mathcal{E} = B\ell v\]

A conductor moving through a magnetic field experiences an induced emf.

0685 / Circuit and Electromagnetic Reference

Magnetic dipole torque

\[\tau = \mu B\sin\theta\]

A magnetic dipole tends to rotate to align with a magnetic field.

0686 / Circuit and Electromagnetic Reference

Magnetic dipole potential energy

\[U = -\boldsymbol{\mu}\cdot\mathbf{B}\]

Potential energy is lowest when the dipole aligns with the field.

0687 / Circuit and Electromagnetic Reference

Self-inductance of a solenoid

\[L = \mu\frac{N^2A}{\ell}\]

Inductance grows with turns squared and cross-sectional area.

0688 / Circuit and Electromagnetic Reference

Inductive time-domain energy change

\[P = LI\frac{dI}{dt}\]

Instantaneous power into an inductor depends on current and the rate of current change.

0689 / Circuit and Electromagnetic Reference

Capacitive reactance

\[X_C = \frac{1}{2\pi fC}\]

Capacitive reactance falls as frequency increases.

0690 / Circuit and Electromagnetic Reference

Inductive reactance

\[X_L = 2\pi fL\]

Inductive reactance rises as frequency increases.

0691 / Circuit and Electromagnetic Reference

Phase angle in an RLC circuit

\[\phi = \tan^{-1}\left(\frac{X_L-X_C}{R}\right)\]

Phase angle measures the lead or lag between voltage and current.

0692 / Circuit and Electromagnetic Reference

Resonant angular frequency

\[\omega_0 = \frac{1}{\sqrt{LC}}\]

At resonance, inductive and capacitive reactance cancel.

0693 / Circuit and Electromagnetic Reference

Q factor of a series RLC circuit

\[Q = \frac{\omega_0 L}{R}\]

Q factor measures how sharp resonance is.

0694 / Circuit and Electromagnetic Reference

Bandwidth of a resonant circuit

\[\Delta f = \frac{f_0}{Q}\]

High-Q circuits have narrow bandwidth.

0695 / Circuit and Electromagnetic Reference

Force on a wire segment vector form

\[\mathbf{F} = I\mathbf{L} \times \mathbf{B}\]

Use the cross product to capture direction and magnitude.

0696 / Circuit and Electromagnetic Reference

Ampere's law

\[\oint \mathbf{B}\cdot d\mathbf{\ell} = \mu_0 I_{enc}\]

Circulating magnetic field depends on enclosed current.

0697 / Circuit and Electromagnetic Reference

Biot-Savart law

\[d\mathbf{B} = \frac{\mu_0}{4\pi}\frac{I\,d\mathbf{\ell} \times \hat{\mathbf{r}}}{r^2}\]

This law builds magnetic fields from current elements.

0698 / Circuit and Electromagnetic Reference

Field at the center of a circular loop

\[B = \frac{\mu_0 I}{2R}\]

A current loop creates a magnetic field through its center.

0699 / Circuit and Electromagnetic Reference

Field on the axis of a circular loop

\[B = \frac{\mu_0 I R^2}{2(R^2+x^2)^{3/2}}\]

Field strength decreases away from the center along the axis.

0700 / Circuit and Electromagnetic Reference

Electric dipole moment

\[p = qd\]

Dipole moment measures charge separation.

0701 / Circuit and Electromagnetic Reference

Dipole potential

\[V = \frac{1}{4\pi\epsilon_0}\frac{p\cos\theta}{r^2}\]

Far from the dipole, electric potential falls with the square of distance.

0702 / Circuit and Electromagnetic Reference

Dipole field on axis

\[E = \frac{1}{4\pi\epsilon_0}\frac{2p}{r^3}\]

Along the dipole axis, field strength falls with the cube of distance.

Chemical Thermodynamics and Solutions

25 formulas

0703 / Chemical Thermodynamics and Solutions

Reaction enthalpy from formation enthalpies

\[\Delta H^\circ_{rxn} = \sum \nu \Delta H_f^\circ(products) - \sum \nu \Delta H_f^\circ(reactants)\]

Use tabulated standard enthalpies of formation to find reaction enthalpy.

0704 / Chemical Thermodynamics and Solutions

Reaction entropy from standard entropies

\[\Delta S^\circ_{rxn} = \sum \nu S^\circ(products) - \sum \nu S^\circ(reactants)\]

Use tabulated standard molar entropies to estimate reaction entropy.

0705 / Chemical Thermodynamics and Solutions

Reaction free energy from formation free energies

\[\Delta G^\circ_{rxn} = \sum \nu \Delta G_f^\circ(products) - \sum \nu \Delta G_f^\circ(reactants)\]

Use standard free energies of formation to estimate spontaneity.

0706 / Chemical Thermodynamics and Solutions

Equilibrium relation between Kp and Kc

\[K_p = K_c(RT)^{\Delta n}\]

Convert between pressure-based and concentration-based equilibrium constants.

0707 / Chemical Thermodynamics and Solutions

Hess's law

\[\Delta H_{overall} = \sum \Delta H_{steps}\]

Reaction enthalpies add when reactions are added.

0708 / Chemical Thermodynamics and Solutions

van't Hoff factor definition

\[i = \frac{observed\ particles}{formula\ units}\]

The van't Hoff factor counts how many dissolved particles each formula unit produces.

0709 / Chemical Thermodynamics and Solutions

Acid-base titration relation

\[M_aV_a\,n_a = M_bV_b\,n_b\]

At equivalence, acid and base moles balance stoichiometrically.

0710 / Chemical Thermodynamics and Solutions

Equivalent weight

\[Equivalent\ Weight = \frac{Molar\ Mass}{n\text{-}factor}\]

Equivalent weight depends on how many protons, electrons, or ions are exchanged.

0711 / Chemical Thermodynamics and Solutions

Normality

\[N = \frac{Equivalents}{Liter}\]

Normality measures equivalent concentration.

0712 / Chemical Thermodynamics and Solutions

Ka-Kb relation

\[K_aK_b = K_w\]

A conjugate acid-base pair satisfies this relation in water.

0713 / Chemical Thermodynamics and Solutions

pKa relation

\[pK_a = -\log K_a\]

pKa is the negative base-10 logarithm of Ka.

0714 / Chemical Thermodynamics and Solutions

pKb relation

\[pK_b = -\log K_b\]

pKb is the negative base-10 logarithm of Kb.

0715 / Chemical Thermodynamics and Solutions

Bohr electron velocity

\[v_n = \frac{Z\alpha c}{n}\]

Hydrogen-like atoms have quantized electron speeds in the Bohr model.

0716 / Chemical Thermodynamics and Solutions

de Broglie wavelength of a particle

\[\lambda = \frac{h}{mv}\]

The wavelength of a moving particle is inversely proportional to momentum.

0717 / Chemical Thermodynamics and Solutions

Molar heat capacity relation

\[C = \frac{q}{n\Delta T}\]

Molar heat capacity uses amount of substance instead of mass.

0718 / Chemical Thermodynamics and Solutions

Entropy change at constant temperature

\[\Delta S = \frac{q_{rev}}{T}\]

Reversible heat transfer divided by absolute temperature gives entropy change.

0719 / Chemical Thermodynamics and Solutions

Reaction quotient from partial pressures

\[Q_p = \frac{P_{products}^{coeff}}{P_{reactants}^{coeff}}\]

Build Qp from partial pressures raised to their stoichiometric powers.

0720 / Chemical Thermodynamics and Solutions

Henry's law

\[C = k_H P\]

Gas solubility in a liquid is proportional to partial pressure above the liquid.

0721 / Chemical Thermodynamics and Solutions

Osmolarity

\[Osmolarity = iM\]

Multiply molarity by the van't Hoff factor to count dissolved particles.

0722 / Chemical Thermodynamics and Solutions

Buffer base form

\[pOH = pK_b + \log\frac{[BH^+]}{[B]}\]

Use this base-form Henderson-Hasselbalch equation for basic buffers.

0723 / Chemical Thermodynamics and Solutions

Gibbs-Helmholtz relation

\[\left(\frac{\partial (\Delta G/T)}{\partial T}\right)_P = -\frac{\Delta H}{T^2}\]

This thermodynamic relation links free-energy change and enthalpy to temperature.

0724 / Chemical Thermodynamics and Solutions

Clapeyron equation

\[\frac{dP}{dT} = \frac{\Delta H}{T\Delta V}\]

This describes the slope of a phase boundary on a pressure-temperature diagram.

0725 / Chemical Thermodynamics and Solutions

van der Waals critical temperature

\[T_c = \frac{8a}{27Rb}\]

The van der Waals constants determine a fluid's critical temperature.

0726 / Chemical Thermodynamics and Solutions

van der Waals critical pressure

\[P_c = \frac{a}{27b^2}\]

The van der Waals constants determine a fluid's critical pressure.

0727 / Chemical Thermodynamics and Solutions

van der Waals critical volume

\[V_c = 3nb\]

The van der Waals constants determine a fluid's critical molar volume.

Astronomy and Biophysics Reference

23 formulas

0728 / Astronomy and Biophysics Reference

Gravitational potential energy of two masses

\[U = -G\frac{m_1m_2}{r}\]

Mutual gravitational potential energy is negative and approaches zero at infinite separation.

0729 / Astronomy and Biophysics Reference

Orbital mechanical energy

\[E = -\frac{GMm}{2a}\]

A bound Keplerian orbit has negative total energy determined by its semi-major axis.

0730 / Astronomy and Biophysics Reference

Vis-viva equation

\[v^2 = GM\left(\frac{2}{r} - \frac{1}{a}\right)\]

Orbital speed depends on the current radius and semi-major axis.

0731 / Astronomy and Biophysics Reference

Schwarzschild radius

\[r_s = \frac{2GM}{c^2}\]

This radius defines the event horizon of a non-rotating black hole.

0732 / Astronomy and Biophysics Reference

Surface escape speed

\[v_e = \sqrt{\frac{2GM}{R}}\]

Escape speed rises with mass and decreases with radius.

0733 / Astronomy and Biophysics Reference

Surface orbital speed

\[v = \sqrt{\frac{GM}{R}}\]

A circular orbit at the surface has this speed in the simplified model.

0734 / Astronomy and Biophysics Reference

Surface flux from luminosity

\[F = \frac{L}{4\pi d^2}\]

Flux at distance d spreads over a sphere of radius d.

0735 / Astronomy and Biophysics Reference

Absolute magnitude relation

\[M = m - 5\log_{10}\left(\frac{d}{10\,pc}\right)\]

Absolute magnitude is the apparent magnitude the object would have at 10 parsecs.

0736 / Astronomy and Biophysics Reference

Parallax distance

\[d\,(pc) = \frac{1}{p\,(arcsec)}\]

Astronomical distance in parsecs is the reciprocal of parallax angle in arcseconds.

0737 / Astronomy and Biophysics Reference

Hubble time estimate

\[t_H \approx \frac{1}{H_0}\]

The inverse of Hubble's constant gives a rough cosmic timescale.

0738 / Astronomy and Biophysics Reference

Body-fat-free mass index

\[FFMI = \frac{Fat\text{-}Free\ Mass}{Height^2}\]

FFMI compares lean body mass to height squared.

0739 / Astronomy and Biophysics Reference

Pulse pressure

\[PP = SBP - DBP\]

Pulse pressure is the difference between systolic and diastolic blood pressure.

0740 / Astronomy and Biophysics Reference

Ejection fraction

\[EF = \frac{Stroke\ Volume}{End\text{-}Diastolic\ Volume} \cdot 100\]

Ejection fraction measures how much ventricular volume is pumped out per beat.

0741 / Astronomy and Biophysics Reference

Cardiac index

\[CI = \frac{Cardiac\ Output}{Body\ Surface\ Area}\]

Cardiac index normalizes cardiac output by body size.

0742 / Astronomy and Biophysics Reference

Respiratory minute volume

\[\dot{V}_E = V_T f\]

Minute ventilation is tidal volume times breathing frequency.

0743 / Astronomy and Biophysics Reference

Oxygen pulse

\[O_2\ Pulse = \frac{\dot{V}_{O_2}}{HR}\]

Oxygen pulse approximates oxygen used per heartbeat.

0744 / Astronomy and Biophysics Reference

Basal metabolic rate surface rule

\[BMR \propto Body\ Surface\ Area\]

Metabolic needs scale roughly with surface area across individuals.

0745 / Astronomy and Biophysics Reference

Drug infusion rate

\[Rate = Concentration \times Flow\]

Infused dose delivery rate equals solution concentration times volumetric flow.

0746 / Astronomy and Biophysics Reference

Creatinine clearance approximation

\[Cl \approx \frac{U\times V}{P}\]

Clearance compares urine concentration times flow to plasma concentration.

0747 / Astronomy and Biophysics Reference

Alveolar gas equation

\[P_AO_2 = P_IO_2 - \frac{P_aCO_2}{R}\]

Estimate alveolar oxygen pressure from inspired oxygen and arterial carbon dioxide.

0748 / Astronomy and Biophysics Reference

Mean corpuscular volume

\[MCV = \frac{Hematocrit \times 10}{RBC\ Count}\]

Mean corpuscular volume estimates average red-blood-cell size.

0749 / Astronomy and Biophysics Reference

Mean corpuscular hemoglobin

\[MCH = \frac{Hemoglobin \times 10}{RBC\ Count}\]

Mean corpuscular hemoglobin estimates average hemoglobin per red blood cell.

0750 / Astronomy and Biophysics Reference

Mean corpuscular hemoglobin concentration

\[MCHC = \frac{Hemoglobin \times 100}{Hematocrit}\]

MCHC estimates hemoglobin concentration within red blood cells.

Interest and Time Value of Money

40 formulas

0751 / Interest and Time Value of Money

Simple interest

\[I = Prt\]

Simple interest grows linearly with principal, rate, and time.

0752 / Interest and Time Value of Money

Total amount with simple interest

\[A = P(1+rt)\]

Add simple interest to principal.

0753 / Interest and Time Value of Money

Compound amount

\[A = P\left(1 + \frac{r}{n}\right)^{nt}\]

Compound growth adds interest on prior interest.

0754 / Interest and Time Value of Money

Continuous compounding

\[A = Pe^{rt}\]

Continuous compounding uses the natural exponential function.

0755 / Interest and Time Value of Money

Future value

\[FV = PV(1+r)^t\]

Future value compounds present money forward in time.

0756 / Interest and Time Value of Money

Present value

\[PV = \frac{FV}{(1+r)^t}\]

Discount future money back to today's value.

0757 / Interest and Time Value of Money

Discount factor

\[DF = \frac{1}{(1+r)^t}\]

A discount factor converts a future amount to present value.

0758 / Interest and Time Value of Money

Effective annual rate

\[EAR = \left(1 + \frac{r}{n}\right)^n - 1\]

EAR converts a nominal quoted rate into an annual compounded rate.

0759 / Interest and Time Value of Money

Nominal rate from EAR

\[r = n\left((1+EAR)^{1/n}-1\right)\]

Recover the nominal rate that compounds n times per year.

0760 / Interest and Time Value of Money

Real rate from nominal rate

\[1+r_{real} = \frac{1+r_{nominal}}{1+\pi}\]

Adjust a nominal return for inflation.

0761 / Interest and Time Value of Money

Approximate real rate

\[r_{real} \approx r_{nominal} - \pi\]

This approximation works well when rates are relatively small.

0762 / Interest and Time Value of Money

Rule of 72

\[Years \approx \frac{72}{Rate\%}\]

Estimate doubling time quickly using 72 divided by the annual rate.

0763 / Interest and Time Value of Money

Doubling time from continuous growth

\[t = \frac{\ln 2}{r}\]

For continuous compounding, the doubling time uses natural logs.

0764 / Interest and Time Value of Money

Single cash-flow net present value

\[NPV = \frac{CF_t}{(1+r)^t}\]

Discount an individual future cash flow to the present.

0765 / Interest and Time Value of Money

Multi-period net present value

\[NPV = \sum_{t=0}^{n} \frac{CF_t}{(1+r)^t}\]

Discount and sum each cash flow across the timeline.

0766 / Interest and Time Value of Money

Perpetuity present value

\[PV = \frac{C}{r}\]

A level perpetuity pays the same amount forever.

0767 / Interest and Time Value of Money

Growing perpetuity present value

\[PV = \frac{C_1}{r-g}\]

Use the next-period cash flow and ensure r is greater than g.

0768 / Interest and Time Value of Money

Ordinary annuity present value

\[PV = PMT\frac{1-(1+r)^{-n}}{r}\]

Discount a level payment stream paid at the end of each period.

0769 / Interest and Time Value of Money

Ordinary annuity future value

\[FV = PMT\frac{(1+r)^n-1}{r}\]

Compound each payment to the end of the annuity.

0770 / Interest and Time Value of Money

Annuity due present value

\[PV_{due} = PV_{ordinary}(1+r)\]

Payments at the beginning of each period are worth one extra period of growth.

0771 / Interest and Time Value of Money

Annuity due future value

\[FV_{due} = FV_{ordinary}(1+r)\]

Payments at the beginning of each period compound one extra period.

0772 / Interest and Time Value of Money

Growing annuity present value

\[PV = PMT\frac{1-\left(\frac{1+g}{1+r}\right)^n}{r-g}\]

Discount a payment stream that grows by g each period.

0773 / Interest and Time Value of Money

Growing annuity future value

\[FV = PMT\frac{(1+r)^n-(1+g)^n}{r-g}\]

Accumulate a payment stream that grows at rate g.

0774 / Interest and Time Value of Money

Future value of a lump sum

\[FV = PV(1+i)^n\]

This is the general periodic compounding formula.

0775 / Interest and Time Value of Money

Present value of a lump sum

\[PV = \frac{FV}{(1+i)^n}\]

This is the general periodic discounting formula.

0776 / Interest and Time Value of Money

Continuous discounting present value

\[PV = FVe^{-rt}\]

Use the exponential discount factor under continuous compounding.

0777 / Interest and Time Value of Money

Equivalent periodic rate

\[i = \left(1+r_{annual}\right)^{1/m}-1\]

Convert an effective annual rate to an equivalent periodic rate.

0778 / Interest and Time Value of Money

Future value with recurring contributions

\[FV = PV(1+i)^n + PMT\frac{(1+i)^n-1}{i}\]

Grow an opening balance and add the future value of contributions.

0779 / Interest and Time Value of Money

Required periodic contribution

\[PMT = \frac{(FV-PV(1+i)^n)i}{(1+i)^n-1}\]

Solve the contribution formula for the recurring payment.

0780 / Interest and Time Value of Money

Internal rate of return condition

\[0 = \sum_{t=0}^{n} \frac{CF_t}{(1+IRR)^t}\]

IRR is the discount rate that makes net present value equal zero.

0781 / Interest and Time Value of Money

Profitability index

\[PI = \frac{PV\ of\ Future\ Cash\ Inflows}{Initial\ Investment}\]

A profitability index above one indicates positive net present value.

0782 / Interest and Time Value of Money

Equivalent annual annuity

\[EAA = NPV\frac{r}{1-(1+r)^{-n}}\]

Convert project value into an equivalent level annual amount.

0783 / Interest and Time Value of Money

Capital recovery factor

\[CRF = \frac{r(1+r)^n}{(1+r)^n-1}\]

Use this to turn a present cost into equal periodic payments.

0784 / Interest and Time Value of Money

Sinking fund factor

\[SFF = \frac{r}{(1+r)^n-1}\]

This gives the periodic deposit required to reach a target future amount.

0785 / Interest and Time Value of Money

Future value interest factor

\[FVIF = (1+r)^n\]

FVIF is the growth multiplier for a single sum.

0786 / Interest and Time Value of Money

Present value interest factor

\[PVIF = \frac{1}{(1+r)^n}\]

PVIF is the discount multiplier for a single future sum.

0787 / Interest and Time Value of Money

Present value interest factor of annuity

\[PVIFA = \frac{1-(1+r)^{-n}}{r}\]

PVIFA is the multiplier for an ordinary annuity present value.

0788 / Interest and Time Value of Money

Future value interest factor of annuity

\[FVIFA = \frac{(1+r)^n-1}{r}\]

FVIFA is the multiplier for an ordinary annuity future value.

0789 / Interest and Time Value of Money

Capital asset discount factor

\[Discount\ Factor_t = \frac{1}{(1+WACC)^t}\]

Use the weighted average cost of capital to discount project cash flows.

0790 / Interest and Time Value of Money

Horizon value

\[HV = \frac{FCF_{n+1}}{r-g}\]

A growing perpetuity can estimate continuing value beyond a forecast horizon.

Loans and Amortization

32 formulas

0791 / Loans and Amortization

Amortized loan payment

\[PMT = P\frac{i(1+i)^n}{(1+i)^n-1}\]

Use this to find a fixed periodic payment on an amortizing loan.

0792 / Loans and Amortization

Loan principal from payment

\[P = PMT\frac{(1+i)^n-1}{i(1+i)^n}\]

Solve the amortization formula for principal.

0793 / Loans and Amortization

Number of payments

\[n = \frac{-\ln\left(1-\frac{iP}{PMT}\right)}{\ln(1+i)}\]

Solve the amortization formula for the payment count.

0794 / Loans and Amortization

Periodic rate from APR

\[i = \frac{APR}{m}\]

Divide the annual percentage rate by periods per year.

0795 / Loans and Amortization

Mortgage principal

\[Principal = Price - Down\ Payment\]

Mortgage amount equals price minus down payment.

0796 / Loans and Amortization

Remaining loan balance

\[B_k = P(1+i)^k - PMT\frac{(1+i)^k-1}{i}\]

This gives balance after k payments.

0797 / Loans and Amortization

Interest portion of a payment

\[Interest_k = iB_{k-1}\]

Interest due equals the periodic rate times the previous balance.

0798 / Loans and Amortization

Principal portion of a payment

\[Principal_k = PMT - Interest_k\]

Subtract the interest from the total payment.

0799 / Loans and Amortization

Total interest paid

\[Total\ Interest = PMT\cdot n - P\]

Total interest is all payments minus original principal.

0800 / Loans and Amortization

Loan-to-value ratio

\[LTV = \frac{Loan\ Amount}{Property\ Value} \cdot 100\]

LTV compares the borrowed amount to collateral value.

0801 / Loans and Amortization

Debt-to-income ratio

\[DTI = \frac{Monthly\ Debt\ Payments}{Gross\ Monthly\ Income} \cdot 100\]

DTI measures debt burden relative to income.

0802 / Loans and Amortization

Break-even refinance time

\[Months = \frac{Closing\ Costs}{Monthly\ Savings}\]

Refinancing pays off after cumulative savings cover upfront costs.

0803 / Loans and Amortization

Biweekly payment equivalent

\[PMT_{biweekly} = \frac{PMT_{monthly}\cdot 12}{26}\]

Convert monthly payments into an equivalent biweekly schedule.

0804 / Loans and Amortization

Simple average daily balance interest

\[Interest = Balance \cdot Daily\ Rate \cdot Days\]

Credit-card interest often uses average daily balance calculations.

0805 / Loans and Amortization

Lease payment approximation

\[Lease\ Payment = Depreciation\ Fee + Finance\ Fee\]

A vehicle lease payment combines depreciation and financing.

0806 / Loans and Amortization

Depreciation fee in a lease

\[Depreciation\ Fee = \frac{Cap\ Cost - Residual}{Lease\ Term}\]

Spread the value loss across the lease term.

0807 / Loans and Amortization

Finance fee in a lease

\[Finance\ Fee = (Cap\ Cost + Residual)\cdot Money\ Factor\]

Lease finance charges depend on capitalized cost, residual, and money factor.

0808 / Loans and Amortization

Money factor to APR

\[APR \approx Money\ Factor \cdot 2400\]

Multiply a lease money factor by 2400 to estimate APR.

0809 / Loans and Amortization

Home affordability ratio

\[Affordable\ Payment = Income \cdot Housing\ Ratio\]

Housing-payment rules often cap payment as a share of income.

0810 / Loans and Amortization

Balloon loan balance

\[B_n = P(1+i)^n - PMT\frac{(1+i)^n-1}{i}\]

A balloon balance is the amount still owed when the loan term ends early.

0811 / Loans and Amortization

Savings goal from monthly deposits

\[PMT = Goal\cdot\frac{i}{(1+i)^n-1}\]

Solve for the monthly deposit required to hit a future target.

0812 / Loans and Amortization

Sinking-fund target value

\[FV = PMT\frac{(1+i)^n-1}{i}\]

Recurring deposits grow into a future fund value.

0813 / Loans and Amortization

APR from finance charge

\[APR \approx \frac{2m\cdot Finance\ Charge}{Principal\cdot(Number\ of\ Payments + 1)}\]

This approximation estimates APR from installment-loan terms.

0814 / Loans and Amortization

Effective monthly payment with taxes and insurance

\[Housing\ Payment = P\&I + Taxes + Insurance + HOA\]

Total monthly housing cost includes escrowed items and association fees.

0815 / Loans and Amortization

Cash flow from rental property

\[Cash\ Flow = Rent - Expenses - Debt\ Service\]

Rental cash flow is income minus operating costs and financing.

0816 / Loans and Amortization

Mortgage constant

\[Mortgage\ Constant = \frac{Annual\ Debt\ Service}{Loan\ Amount}\]

This expresses annual loan cost as a percentage of original loan amount.

0817 / Loans and Amortization

Debt service coverage ratio

\[DSCR = \frac{NOI}{Debt\ Service}\]

Lenders use DSCR to assess whether income covers loan payments.

0818 / Loans and Amortization

Price per square foot

\[Price\ per\ SqFt = \frac{Price}{Area}\]

Normalize property prices by floor area.

0819 / Loans and Amortization

Amortization factor

\[Factor = \frac{i(1+i)^n}{(1+i)^n-1}\]

Multiply this factor by principal to get the periodic payment.

0820 / Loans and Amortization

Home equity

\[Equity = Property\ Value - Loan\ Balance\]

Equity is the owner's value in the property after subtracting debt.

0821 / Loans and Amortization

Cash-out refinance proceeds

\[Cash\ Out = New\ Loan - Old\ Balance - Closing\ Costs\]

This estimates cash received from a refinance after paying off the prior loan.

0822 / Loans and Amortization

Break-even occupancy

\[Occupancy = \frac{Fixed\ Costs}{Average\ Rate - Variable\ Cost\ per\ Unit}\]

Solve for the occupancy or unit count required to cover costs.

Investing and Portfolio Theory

43 formulas

0823 / Investing and Portfolio Theory

Return on investment

\[ROI = \frac{Gain - Cost}{Cost} \cdot 100\]

ROI compares profit to the original cost.

0824 / Investing and Portfolio Theory

Holding period return

\[HPR = \frac{Income + Ending\ Value - Beginning\ Value}{Beginning\ Value}\]

Include both price change and income received.

0825 / Investing and Portfolio Theory

Compound annual growth rate

\[CAGR = \left(\frac{Ending}{Beginning}\right)^{1/n} - 1\]

CAGR smooths multi-year growth into an annualized rate.

0826 / Investing and Portfolio Theory

Dividend yield

\[Dividend\ Yield = \frac{Annual\ Dividend\ per\ Share}{Price\ per\ Share}\]

Dividend yield expresses income relative to price.

0827 / Investing and Portfolio Theory

Dividend payout ratio

\[Payout\ Ratio = \frac{Dividends}{Net\ Income}\]

This measures how much profit is distributed to shareholders.

0828 / Investing and Portfolio Theory

Retention ratio

\[Retention\ Ratio = 1 - Payout\ Ratio\]

The retention ratio measures what share of earnings stays in the business.

0829 / Investing and Portfolio Theory

Expected portfolio return

\[E(R_p) = \sum w_i E(R_i)\]

Portfolio expected return is a weighted average of asset returns.

0830 / Investing and Portfolio Theory

Portfolio variance

\[\sigma_p^2 = \sum_i\sum_j w_i w_j \operatorname{Cov}(R_i,R_j)\]

Portfolio variance depends on weights, asset variances, and covariances.

0831 / Investing and Portfolio Theory

Two-asset portfolio variance

\[\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\sigma_1\sigma_2\rho_{12}\]

Use correlation to simplify the covariance term.

0832 / Investing and Portfolio Theory

Portfolio standard deviation

\[\sigma_p = \sqrt{\sigma_p^2}\]

Take the square root of portfolio variance to get volatility.

0833 / Investing and Portfolio Theory

Sharpe ratio

\[Sharpe = \frac{R_p - R_f}{\sigma_p}\]

Sharpe ratio measures excess return per unit of total risk.

0834 / Investing and Portfolio Theory

Treynor ratio

\[Treynor = \frac{R_p - R_f}{\beta_p}\]

Treynor ratio measures excess return per unit of systematic risk.

0835 / Investing and Portfolio Theory

Jensen's alpha

\[\alpha = R_p - [R_f + \beta_p(R_m - R_f)]\]

Alpha compares realized return to the return predicted by CAPM.

0836 / Investing and Portfolio Theory

CAPM expected return

\[E(R_i) = R_f + \beta_i(E(R_m)-R_f)\]

CAPM links expected return to systematic market risk.

0837 / Investing and Portfolio Theory

Beta from covariance

\[\beta_i = \frac{\operatorname{Cov}(R_i,R_m)}{\operatorname{Var}(R_m)}\]

Beta compares asset sensitivity to market variance.

0838 / Investing and Portfolio Theory

Security market line

\[E(R) = R_f + \beta(E(R_m)-R_f)\]

The security market line is the CAPM relationship graphed.

0839 / Investing and Portfolio Theory

Price-to-earnings ratio

\[P/E = \frac{Price\ per\ Share}{Earnings\ per\ Share}\]

P/E compares market price to earnings power.

0840 / Investing and Portfolio Theory

Earnings per share

\[EPS = \frac{Net\ Income - Preferred\ Dividends}{Weighted\ Average\ Shares}\]

EPS distributes earnings across outstanding common shares.

0841 / Investing and Portfolio Theory

Book value per share

\[BVPS = \frac{Common\ Equity}{Shares\ Outstanding}\]

Book value per share compares net assets to share count.

0842 / Investing and Portfolio Theory

Price-to-book ratio

\[P/B = \frac{Market\ Price\ per\ Share}{Book\ Value\ per\ Share}\]

P/B compares market valuation to accounting book value.

0843 / Investing and Portfolio Theory

Enterprise value

\[EV = Market\ Cap + Debt - Cash\]

Enterprise value estimates total operating business value.

0844 / Investing and Portfolio Theory

EV to EBITDA

\[EV/EBITDA = \frac{Enterprise\ Value}{EBITDA}\]

This multiple compares business value to cash operating profit.

0845 / Investing and Portfolio Theory

Free cash flow to firm

\[FCFF = EBIT(1-T) + Depreciation - CapEx - \Delta NWC\]

FCFF measures cash available to all capital providers.

0846 / Investing and Portfolio Theory

Free cash flow to equity

\[FCFE = Net\ Income + Depreciation - CapEx - \Delta NWC + Net\ Borrowing\]

FCFE measures cash available to equity holders.

0847 / Investing and Portfolio Theory

Market capitalization

\[Market\ Cap = Share\ Price \times Shares\ Outstanding\]

Market cap is the total market value of equity.

0848 / Investing and Portfolio Theory

Dividend discount model

\[P_0 = \frac{D_1}{r-g}\]

Use the Gordon growth model for a stock with stable perpetual dividend growth.

0849 / Investing and Portfolio Theory

Required return from dividend model

\[r = \frac{D_1}{P_0} + g\]

Rearrange the Gordon growth model to solve for required return.

0850 / Investing and Portfolio Theory

Growth rate from ROE and retention

\[g = ROE \times Retention\ Ratio\]

Sustainable growth ties profitability to retained earnings.

0851 / Investing and Portfolio Theory

Current yield

\[Current\ Yield = \frac{Annual\ Coupon}{Bond\ Price}\]

Current yield measures coupon income relative to bond price.

0852 / Investing and Portfolio Theory

Bond price

\[P = \sum_{t=1}^{n}\frac{C}{(1+y)^t} + \frac{F}{(1+y)^n}\]

Bond value is the present value of coupons plus face value.

0853 / Investing and Portfolio Theory

Zero-coupon bond price

\[P = \frac{F}{(1+y)^n}\]

A zero-coupon bond has no periodic coupons.

0854 / Investing and Portfolio Theory

Approximate yield to maturity

\[YTM \approx \frac{C + \frac{F-P}{n}}{\frac{F+P}{2}}\]

This gives a quick estimate of a bond's yield.

0855 / Investing and Portfolio Theory

Duration approximation

\[D \approx \frac{\Delta P / P}{\Delta y}\]

Duration estimates price sensitivity to yield changes.

0856 / Investing and Portfolio Theory

Modified duration

\[D_{mod} = \frac{D_{Mac}}{1+y}\]

Modified duration scales Macaulay duration for price sensitivity.

0857 / Investing and Portfolio Theory

Current ratio of return and risk

\[Utility = E(R) - \frac{1}{2}A\sigma^2\]

Mean-variance utility trades expected return against risk aversion.

0858 / Investing and Portfolio Theory

Kelly criterion

\[f^* = \frac{bp - q}{b}\]

The Kelly fraction estimates an optimal betting fraction under specific assumptions.

0859 / Investing and Portfolio Theory

Maximum drawdown

\[MDD = \frac{Trough - Peak}{Peak}\]

Maximum drawdown measures the worst peak-to-trough decline.

0860 / Investing and Portfolio Theory

Calmar ratio

\[Calmar = \frac{Annualized\ Return}{|MDD|}\]

The Calmar ratio compares return to worst drawdown.

0861 / Investing and Portfolio Theory

Sortino ratio

\[Sortino = \frac{R_p - R_f}{\sigma_d}\]

Sortino uses downside deviation instead of total volatility.

0862 / Investing and Portfolio Theory

Downside deviation

\[\sigma_d = \sqrt{\frac{\sum \min(0,R_i - MAR)^2}{n}}\]

Downside deviation tracks only shortfalls below a target or minimum acceptable return.

0863 / Investing and Portfolio Theory

Expense ratio drag

\[Net\ Return \approx Gross\ Return - Expense\ Ratio\]

Fund fees reduce investor returns over time.

0864 / Investing and Portfolio Theory

Tax-equivalent yield

\[TEY = \frac{Tax\ Free\ Yield}{1-Tax\ Rate}\]

Compare municipal bond yields to taxable alternatives.

0865 / Investing and Portfolio Theory

After-tax return

\[After\ Tax\ Return = Pre\ Tax\ Return\times(1-Tax\ Rate)\]

Adjust investment performance for taxes.

Accounting and Business Ratios

59 formulas

0866 / Accounting and Business Ratios

Gross profit

\[Gross\ Profit = Revenue - COGS\]

Gross profit subtracts cost of goods sold from revenue.

0867 / Accounting and Business Ratios

Gross margin

\[Gross\ Margin = \frac{Gross\ Profit}{Revenue} \cdot 100\]

Gross margin shows the share of sales left after direct product costs.

0868 / Accounting and Business Ratios

Operating income

\[Operating\ Income = Gross\ Profit - Operating\ Expenses\]

Operating income measures core business profit before interest and taxes.

0869 / Accounting and Business Ratios

Operating margin

\[Operating\ Margin = \frac{Operating\ Income}{Revenue} \cdot 100\]

Operating margin expresses operating profit as a share of revenue.

0870 / Accounting and Business Ratios

Net margin

\[Net\ Margin = \frac{Net\ Income}{Revenue} \cdot 100\]

Net margin shows how much of each sales dollar becomes profit.

0871 / Accounting and Business Ratios

EBITDA

\[EBITDA = EBIT + Depreciation + Amortization\]

EBITDA adds noncash depreciation and amortization back to operating profit.

0872 / Accounting and Business Ratios

EBITDA margin

\[EBITDA\ Margin = \frac{EBITDA}{Revenue} \cdot 100\]

EBITDA margin compares cash-like operating profit to revenue.

0873 / Accounting and Business Ratios

Current ratio

\[Current\ Ratio = \frac{Current\ Assets}{Current\ Liabilities}\]

Current ratio measures short-term liquidity.

0874 / Accounting and Business Ratios

Quick ratio

\[Quick\ Ratio = \frac{Current\ Assets - Inventory}{Current\ Liabilities}\]

Quick ratio excludes inventory from current assets.

0875 / Accounting and Business Ratios

Cash ratio

\[Cash\ Ratio = \frac{Cash + Marketable\ Securities}{Current\ Liabilities}\]

Cash ratio focuses on the most liquid current assets.

0876 / Accounting and Business Ratios

Working capital

\[Working\ Capital = Current\ Assets - Current\ Liabilities\]

Working capital is the dollar buffer for short-term obligations.

0877 / Accounting and Business Ratios

Inventory turnover

\[Inventory\ Turnover = \frac{COGS}{Average\ Inventory}\]

Inventory turnover measures how quickly stock is sold and replaced.

0878 / Accounting and Business Ratios

Days inventory outstanding

\[DIO = \frac{365}{Inventory\ Turnover}\]

DIO measures how many days inventory stays on hand.

0879 / Accounting and Business Ratios

Receivables turnover

\[Receivables\ Turnover = \frac{Credit\ Sales}{Average\ Accounts\ Receivable}\]

Receivables turnover measures how quickly customers pay.

0880 / Accounting and Business Ratios

Days sales outstanding

\[DSO = \frac{365}{Receivables\ Turnover}\]

DSO measures how many days sales remain uncollected.

0881 / Accounting and Business Ratios

Payables turnover

\[Payables\ Turnover = \frac{COGS}{Average\ Accounts\ Payable}\]

Payables turnover measures how quickly the business pays suppliers.

0882 / Accounting and Business Ratios

Days payable outstanding

\[DPO = \frac{365}{Payables\ Turnover}\]

DPO measures how long payables stay unpaid.

0883 / Accounting and Business Ratios

Cash conversion cycle

\[CCC = DIO + DSO - DPO\]

The cash conversion cycle estimates how long cash is tied up in operations.

0884 / Accounting and Business Ratios

Asset turnover

\[Asset\ Turnover = \frac{Revenue}{Average\ Total\ Assets}\]

Asset turnover measures revenue generated per dollar of assets.

0885 / Accounting and Business Ratios

Fixed-asset turnover

\[Fixed\ Asset\ Turnover = \frac{Revenue}{Average\ Net\ Fixed\ Assets}\]

This ratio focuses on the productivity of long-lived assets.

0886 / Accounting and Business Ratios

Return on assets

\[ROA = \frac{Net\ Income}{Average\ Total\ Assets} \cdot 100\]

ROA measures profit produced by total assets.

0887 / Accounting and Business Ratios

Return on equity

\[ROE = \frac{Net\ Income}{Average\ Equity} \cdot 100\]

ROE measures profit generated for owners.

0888 / Accounting and Business Ratios

Return on invested capital

\[ROIC = \frac{NOPAT}{Invested\ Capital} \cdot 100\]

ROIC compares after-tax operating profit to capital invested in the business.

0889 / Accounting and Business Ratios

Debt ratio

\[Debt\ Ratio = \frac{Total\ Debt}{Total\ Assets}\]

Debt ratio compares total debt to total assets.

0890 / Accounting and Business Ratios

Debt-to-equity ratio

\[D/E = \frac{Total\ Debt}{Total\ Equity}\]

Debt-to-equity compares creditor financing to owner financing.

0891 / Accounting and Business Ratios

Debt-to-capital ratio

\[Debt\ to\ Capital = \frac{Debt}{Debt + Equity}\]

This ratio shows what share of capital structure is debt.

0892 / Accounting and Business Ratios

Interest coverage ratio

\[Interest\ Coverage = \frac{EBIT}{Interest\ Expense}\]

Interest coverage measures how many times operating profit covers interest.

0893 / Accounting and Business Ratios

Fixed-charge coverage ratio

\[Fixed\ Charge\ Coverage = \frac{EBIT + Fixed\ Charges\ Before\ Interest\ and\ Tax}{Interest + Fixed\ Charges}\]

This ratio broadens interest coverage to include other fixed obligations.

0894 / Accounting and Business Ratios

Contribution margin

\[Contribution\ Margin = Sales - Variable\ Costs\]

Contribution margin measures what remains to cover fixed costs and profit.

0895 / Accounting and Business Ratios

Contribution margin ratio

\[CMR = \frac{Contribution\ Margin}{Sales}\]

The contribution margin ratio expresses contribution margin as a share of sales.

0896 / Accounting and Business Ratios

Break-even units

\[Break\text{-}Even\ Units = \frac{Fixed\ Costs}{Selling\ Price - Variable\ Cost\ per\ Unit}\]

Break-even units show how many units must be sold to cover fixed costs.

0897 / Accounting and Business Ratios

Break-even sales

\[Break\text{-}Even\ Sales = \frac{Fixed\ Costs}{Contribution\ Margin\ Ratio}\]

Break-even sales use contribution margin ratio instead of unit contribution.

0898 / Accounting and Business Ratios

Operating leverage

\[DOL = \frac{Contribution\ Margin}{Operating\ Income}\]

Operating leverage measures earnings sensitivity to changes in sales.

0899 / Accounting and Business Ratios

Net working capital ratio

\[NWC\ Ratio = \frac{Current\ Assets - Current\ Liabilities}{Total\ Assets}\]

This ratio scales working capital by total assets.

0900 / Accounting and Business Ratios

Book value

\[Book\ Value = Total\ Assets - Total\ Liabilities\]

Book value equals owners' equity on the balance sheet.

0901 / Accounting and Business Ratios

Altman Z-score

\[Z = 1.2X_1 + 1.4X_2 + 3.3X_3 + 0.6X_4 + 1.0X_5\]

Altman Z-score combines liquidity, profitability, leverage, and activity measures to estimate bankruptcy risk.

0902 / Accounting and Business Ratios

Accounts receivable percentage of sales

\[AR\%\ of\ Sales = \frac{Accounts\ Receivable}{Revenue} \cdot 100\]

This ratio compares outstanding receivables to sales.

0903 / Accounting and Business Ratios

Inventory percentage of assets

\[Inventory\%\ of\ Assets = \frac{Inventory}{Total\ Assets} \cdot 100\]

This ratio shows how much of the asset base is tied up in inventory.

0904 / Accounting and Business Ratios

SG&A ratio

\[SG\&A\ Ratio = \frac{SG\&A}{Revenue} \cdot 100\]

This ratio compares overhead spending to sales.

0905 / Accounting and Business Ratios

CapEx ratio

\[CapEx\ Ratio = \frac{Capital\ Expenditures}{Revenue} \cdot 100\]

This ratio compares capital spending to revenue.

0906 / Accounting and Business Ratios

Operating cash flow ratio

\[Operating\ Cash\ Flow\ Ratio = \frac{Operating\ Cash\ Flow}{Current\ Liabilities}\]

This cash-based ratio measures short-term debt coverage.

0907 / Accounting and Business Ratios

Cash flow margin

\[Cash\ Flow\ Margin = \frac{Operating\ Cash\ Flow}{Revenue} \cdot 100\]

Cash flow margin shows cash generation per dollar of sales.

0908 / Accounting and Business Ratios

Free cash flow margin

\[FCF\ Margin = \frac{Free\ Cash\ Flow}{Revenue} \cdot 100\]

This ratio shows the share of revenue that becomes free cash flow.

0909 / Accounting and Business Ratios

Asset-to-equity ratio

\[Asset\ to\ Equity = \frac{Total\ Assets}{Total\ Equity}\]

This ratio is a leverage multiple used in DuPont analysis.

0910 / Accounting and Business Ratios

Equity multiplier

\[Equity\ Multiplier = \frac{Average\ Assets}{Average\ Equity}\]

Equity multiplier shows the contribution of leverage to ROE.

0911 / Accounting and Business Ratios

DuPont formula

\[ROE = Net\ Margin \times Asset\ Turnover \times Equity\ Multiplier\]

DuPont analysis breaks return on equity into profitability, efficiency, and leverage.

0912 / Accounting and Business Ratios

Sustainable growth rate

\[SGR = ROE \times Retention\ Ratio\]

Sustainable growth estimates how fast sales and assets can grow without new external equity.

0913 / Accounting and Business Ratios

Burn rate

\[Burn\ Rate = \frac{Cash\ Used}{Months}\]

Burn rate shows how quickly a company is spending cash.

0914 / Accounting and Business Ratios

Runway

\[Runway = \frac{Cash\ Balance}{Monthly\ Burn}\]

Runway estimates how many months a company can operate before cash runs out.

0915 / Accounting and Business Ratios

Customer acquisition cost

\[CAC = \frac{Sales\ and\ Marketing\ Spend}{New\ Customers}\]

CAC measures the average cost to acquire one new customer.

0916 / Accounting and Business Ratios

Customer lifetime value

\[LTV = ARPU \times Gross\ Margin \times Average\ Customer\ Lifespan\]

LTV estimates total gross profit from a typical customer over its life.

0917 / Accounting and Business Ratios

LTV-to-CAC ratio

\[LTV:CAC = \frac{LTV}{CAC}\]

Compare customer value to acquisition cost.

0918 / Accounting and Business Ratios

Average revenue per user

\[ARPU = \frac{Revenue}{Average\ Users}\]

ARPU measures monetization per user.

0919 / Accounting and Business Ratios

Monthly recurring revenue

\[MRR = Subscribers \times Average\ Monthly\ Price\]

MRR measures predictable monthly subscription revenue.

0920 / Accounting and Business Ratios

Annual recurring revenue

\[ARR = 12 \times MRR\]

ARR annualizes recurring subscription revenue.

0921 / Accounting and Business Ratios

Churn rate

\[Churn = \frac{Customers\ Lost}{Customers\ at\ Start} \cdot 100\]

Churn measures the share of customers that leave.

0922 / Accounting and Business Ratios

Retention rate

\[Retention = 1 - Churn\]

Retention is the share of customers that remain.

0923 / Accounting and Business Ratios

Rule of 40

\[Rule\ of\ 40 = Revenue\ Growth\ Rate + EBITDA\ Margin\]

Software businesses often target a combined growth-plus-margin score above 40.

0924 / Accounting and Business Ratios

Burn multiple

\[Burn\ Multiple = \frac{Net\ Burn}{Net\ New\ ARR}\]

Burn multiple compares cash burn to recurring revenue added.

Valuation, Real Estate, and Tax

30 formulas

0925 / Valuation, Real Estate, and Tax

Net operating income

\[NOI = Revenue - Operating\ Expenses\]

NOI excludes financing, income taxes, and capital expenditures.

0926 / Valuation, Real Estate, and Tax

Capitalization rate

\[Cap\ Rate = \frac{NOI}{Property\ Value}\]

Cap rate compares income to asset value.

0927 / Valuation, Real Estate, and Tax

Cash-on-cash return

\[Cash\text{-}on\text{-}Cash = \frac{Annual\ Pre\text{-}Tax\ Cash\ Flow}{Cash\ Invested} \cdot 100\]

This compares cash received to cash invested.

0928 / Valuation, Real Estate, and Tax

Gross rent multiplier

\[GRM = \frac{Property\ Price}{Gross\ Annual\ Rent}\]

GRM is a quick property-screening multiple.

0929 / Valuation, Real Estate, and Tax

Operating expense ratio

\[OER = \frac{Operating\ Expenses}{Gross\ Operating\ Income}\]

Operating expense ratio compares operating costs to gross property income.

0930 / Valuation, Real Estate, and Tax

Debt yield

\[Debt\ Yield = \frac{NOI}{Loan\ Amount}\]

Debt yield compares property cash generation directly to debt size.

0931 / Valuation, Real Estate, and Tax

Break-even ratio

\[Break\text{-}Even\ Ratio = \frac{Operating\ Expenses + Debt\ Service}{Gross\ Operating\ Income}\]

This ratio shows how much income is required to cover property costs.

0932 / Valuation, Real Estate, and Tax

Internal growth rate

\[IGR = \frac{ROA \times Retention}{1 - ROA \times Retention}\]

Internal growth rate estimates maximum growth without external financing.

0933 / Valuation, Real Estate, and Tax

Weighted average cost of capital

\[WACC = \frac{E}{V}R_e + \frac{D}{V}R_d(1-T)\]

WACC blends equity and after-tax debt costs.

0934 / Valuation, Real Estate, and Tax

Cost of equity from CAPM

\[R_e = R_f + \beta(R_m - R_f)\]

Use CAPM to estimate shareholder required return.

0935 / Valuation, Real Estate, and Tax

After-tax cost of debt

\[R_d(1-T)\]

Interest tax shields lower the effective cost of debt.

0936 / Valuation, Real Estate, and Tax

Terminal value with perpetual growth

\[TV = \frac{FCF_{n+1}}{WACC-g}\]

DCF models often use a perpetuity to estimate post-forecast value.

0937 / Valuation, Real Estate, and Tax

Terminal value with exit multiple

\[TV = EBITDA_n \times Exit\ Multiple\]

An exit multiple approach estimates terminal value from comparable market multiples.

0938 / Valuation, Real Estate, and Tax

Tax shield

\[Tax\ Shield = Deduction \times Tax\ Rate\]

Tax shields measure the tax savings from deductible expenses.

0939 / Valuation, Real Estate, and Tax

Depreciation tax shield

\[Depreciation\ Tax\ Shield = Depreciation \times Tax\ Rate\]

Depreciation reduces taxable income and creates a tax benefit.

0940 / Valuation, Real Estate, and Tax

Present value of a tax shield

\[PV(Tax\ Shield) = \sum \frac{Tax\ Shield_t}{(1+r)^t}\]

Discount future tax savings to the present.

0941 / Valuation, Real Estate, and Tax

After-tax operating income

\[NOPAT = EBIT(1-T)\]

NOPAT strips out financing effects while accounting for taxes.

0942 / Valuation, Real Estate, and Tax

Economic value added

\[EVA = NOPAT - (Invested\ Capital \times WACC)\]

EVA measures whether returns exceed capital costs.

0943 / Valuation, Real Estate, and Tax

Residual income

\[RI = Net\ Income - Equity\ Charge\]

Residual income compares profit to the required return on book equity.

0944 / Valuation, Real Estate, and Tax

Market value added

\[MVA = Market\ Value - Invested\ Capital\]

MVA compares market value to capital invested in the business.

0945 / Valuation, Real Estate, and Tax

Price-to-sales ratio

\[P/S = \frac{Market\ Cap}{Revenue}\]

P/S compares equity value to revenue.

0946 / Valuation, Real Estate, and Tax

Enterprise value to sales

\[EV/Sales = \frac{Enterprise\ Value}{Revenue}\]

This multiple compares total operating value to revenue.

0947 / Valuation, Real Estate, and Tax

Price-to-cash-flow ratio

\[P/CF = \frac{Share\ Price}{Cash\ Flow\ per\ Share}\]

This ratio compares market price to operating cash generation.

0948 / Valuation, Real Estate, and Tax

After-tax yield

\[After\text{-}Tax\ Yield = Pre\text{-}Tax\ Yield\times(1-T)\]

Taxable yields shrink after income taxes are applied.

0949 / Valuation, Real Estate, and Tax

Nominal GDP growth approximation

\[Nominal\ Growth \approx Real\ Growth + Inflation\]

Nominal growth combines real expansion with price-level changes.

0950 / Valuation, Real Estate, and Tax

Price elasticity of demand

\[E_d = \frac{\%\ Change\ in\ Quantity}{\%\ Change\ in\ Price}\]

Elasticity measures how sensitive quantity is to price changes.

0951 / Valuation, Real Estate, and Tax

Contribution margin from price and variable cost

\[Unit\ CM = Price - Variable\ Cost\]

Unit contribution shows what one sale adds toward fixed costs and profit.

0952 / Valuation, Real Estate, and Tax

Net present value of tax savings from depreciation

\[NPV = \sum \frac{Depreciation_t \times Tax\ Rate}{(1+r)^t}\]

Discount each year's depreciation tax shield to present value.

0953 / Valuation, Real Estate, and Tax

Leverage ratio for real estate

\[Leverage = \frac{Loan\ Amount}{Equity\ Invested}\]

Leverage compares borrowed capital to equity capital.

0954 / Valuation, Real Estate, and Tax

Equity build-up return

\[Equity\ Build\text{-}Up\ Return = \frac{Principal\ Paid\ Down}{Equity\ Invested}\]

Part of real-estate return comes from reducing loan balance over time.

Corporate Finance and Valuation Reference

25 formulas

0955 / Corporate Finance and Valuation Reference

Net present value from free cash flow

\[Enterprise\ Value = \sum_{t=1}^{n}\frac{FCFF_t}{(1+WACC)^t} + \frac{TV}{(1+WACC)^n}\]

Discount free cash flow to the firm and terminal value at the weighted average cost of capital.

0956 / Corporate Finance and Valuation Reference

Equity value from enterprise value

\[Equity\ Value = Enterprise\ Value - Debt + Cash\]

Convert total operating value into value attributable to equity holders.

0957 / Corporate Finance and Valuation Reference

Preferred stock cost

\[R_p = \frac{D_p}{P_p}\]

Preferred stock cost equals the dividend divided by current preferred price.

0958 / Corporate Finance and Valuation Reference

Gordon growth cost of equity

\[R_e = \frac{D_1}{P_0} + g\]

A dividend-growth model can estimate required return on equity.

0959 / Corporate Finance and Valuation Reference

Unlevered beta

\[\beta_U = \frac{\beta_L}{1 + (1-T)\frac{D}{E}}\]

Remove capital-structure effects from an observed levered beta.

0960 / Corporate Finance and Valuation Reference

Levered beta

\[\beta_L = \beta_U\left[1 + (1-T)\frac{D}{E}\right]\]

Add capital-structure effects back to an unlevered beta.

0961 / Corporate Finance and Valuation Reference

Market-to-book ratio

\[M/B = \frac{Market\ Value\ of\ Equity}{Book\ Value\ of\ Equity}\]

Compare market value to balance-sheet equity.

0962 / Corporate Finance and Valuation Reference

PEG ratio

\[PEG = \frac{P/E}{Earnings\ Growth\ Rate}\]

PEG adjusts a price-to-earnings multiple for growth.

0963 / Corporate Finance and Valuation Reference

EV to EBIT

\[EV/EBIT = \frac{Enterprise\ Value}{EBIT}\]

Compare total business value to operating profit before interest and tax.

0964 / Corporate Finance and Valuation Reference

EV to FCFF

\[EV/FCFF = \frac{Enterprise\ Value}{FCFF}\]

This valuation multiple compares business value to free cash flow to the firm.

0965 / Corporate Finance and Valuation Reference

Price-to-free-cash-flow ratio

\[P/FCF = \frac{Market\ Cap}{Free\ Cash\ Flow}\]

Compare equity valuation to free cash flow generation.

0966 / Corporate Finance and Valuation Reference

Dividend coverage ratio

\[Dividend\ Coverage = \frac{EPS}{Dividend\ per\ Share}\]

Dividend coverage shows how many times earnings cover dividends.

0967 / Corporate Finance and Valuation Reference

Interest-bearing debt

\[Debt = Short\text{-}Term\ Debt + Long\text{-}Term\ Debt\]

Total interest-bearing debt combines current and noncurrent borrowings.

0968 / Corporate Finance and Valuation Reference

Net debt

\[Net\ Debt = Debt - Cash\]

Net debt offsets debt with available cash balances.

0969 / Corporate Finance and Valuation Reference

Debt to EBITDA ratio

\[Debt/EBITDA = \frac{Debt}{EBITDA}\]

This leverage ratio compares debt load to cash operating profit.

0970 / Corporate Finance and Valuation Reference

Net debt to EBITDA ratio

\[Net\ Debt/EBITDA = \frac{Debt - Cash}{EBITDA}\]

Offset debt with cash before comparing to EBITDA.

0971 / Corporate Finance and Valuation Reference

Plowback ratio

\[Plowback = \frac{Retained\ Earnings}{Net\ Income}\]

The plowback ratio equals the share of earnings not paid as dividends.

0972 / Corporate Finance and Valuation Reference

Weighted average shares outstanding

\[WASO = \sum Shares_i \times Time\ Weight_i\]

Use time weighting when share count changes during the period.

0973 / Corporate Finance and Valuation Reference

Diluted EPS treasury-stock method

\[Incremental\ Shares = Options - \frac{Options \times Exercise\ Price}{Average\ Market\ Price}\]

The treasury-stock method estimates dilutive shares from options and warrants.

0974 / Corporate Finance and Valuation Reference

Book debt ratio

\[Book\ Debt\ Ratio = \frac{Book\ Debt}{Book\ Debt + Book\ Equity}\]

Use balance-sheet values to measure financing mix.

0975 / Corporate Finance and Valuation Reference

Market debt ratio

\[Market\ Debt\ Ratio = \frac{Debt}{Debt + Market\ Value\ of\ Equity}\]

Use market equity value when modeling capital structure.

0976 / Corporate Finance and Valuation Reference

Unlevered free cash flow

\[UFCF = EBIT(1-T) + D\&A - CapEx - \Delta NWC\]

Unlevered free cash flow matches free cash flow to the firm.

0977 / Corporate Finance and Valuation Reference

Levered free cash flow

\[LFCF = CFO - CapEx + Net\ Borrowing\]

Levered free cash flow measures cash available after debt financing.

0978 / Corporate Finance and Valuation Reference

Equity charge

\[Equity\ Charge = Equity\ Capital \times Cost\ of\ Equity\]

Residual-income models subtract the required return on equity capital.

0979 / Corporate Finance and Valuation Reference

Residual value from residual income

\[Value = Book\ Value + \sum \frac{RI_t}{(1+r)^t}\]

Residual-income valuation adds discounted residual income to current book value.

Personal Finance and Tax Planning

21 formulas

0980 / Personal Finance and Tax Planning

Net worth

\[Net\ Worth = Total\ Assets - Total\ Liabilities\]

Net worth measures what remains after paying off all liabilities.

0981 / Personal Finance and Tax Planning

Savings rate

\[Savings\ Rate = \frac{Savings}{Income} \cdot 100\]

Savings rate compares money set aside to income earned.

0982 / Personal Finance and Tax Planning

Expense rate

\[Expense\ Rate = \frac{Expenses}{Income} \cdot 100\]

Expense rate compares spending to income.

0983 / Personal Finance and Tax Planning

Emergency fund months

\[Months\ of\ Buffer = \frac{Cash\ Reserves}{Monthly\ Essential\ Expenses}\]

This shows how many months of core expenses cash savings can cover.

0984 / Personal Finance and Tax Planning

50/30/20 needs bucket

\[Needs = Income \times 0.50\]

A common budgeting rule allocates 50 percent of after-tax income to needs.

0985 / Personal Finance and Tax Planning

50/30/20 wants bucket

\[Wants = Income \times 0.30\]

A common budgeting rule allocates 30 percent of after-tax income to wants.

0986 / Personal Finance and Tax Planning

50/30/20 savings bucket

\[Savings = Income \times 0.20\]

A common budgeting rule allocates 20 percent of after-tax income to saving or debt payoff.

0987 / Personal Finance and Tax Planning

Safe annual withdrawal

\[Withdrawal = Portfolio \times Withdrawal\ Rate\]

A safe-withdrawal estimate converts portfolio value into a possible annual spending amount.

0988 / Personal Finance and Tax Planning

Years of retirement funding at constant withdrawal

\[Years = \frac{Portfolio}{Annual\ Withdrawal}\]

This simple estimate ignores investment growth and inflation.

0989 / Personal Finance and Tax Planning

Future college cost with inflation

\[Future\ Cost = Current\ Cost(1+i)^n\]

Inflation compounds education costs forward over time.

0990 / Personal Finance and Tax Planning

Required retirement nest egg

\[Nest\ Egg = \frac{Annual\ Spending}{Withdrawal\ Rate}\]

Invert the withdrawal rule to estimate the needed portfolio size.

0991 / Personal Finance and Tax Planning

After-tax income

\[After\text{-}Tax\ Income = Gross\ Income - Taxes\]

After-tax income is the amount available to spend or save.

0992 / Personal Finance and Tax Planning

Effective tax rate

\[Effective\ Tax\ Rate = \frac{Total\ Tax}{Taxable\ Income} \cdot 100\]

Effective tax rate compares total tax paid to taxable income.

0993 / Personal Finance and Tax Planning

Marginal tax amount on extra income

\[Tax\ on\ Extra\ Income = Extra\ Income \times Marginal\ Rate\]

Marginal tax focuses on the next dollars earned.

0994 / Personal Finance and Tax Planning

Taxable income

\[Taxable\ Income = Gross\ Income - Deductions - Exemptions\]

Taxable income is the income base used to calculate tax.

0995 / Personal Finance and Tax Planning

After-tax investment value

\[After\text{-}Tax\ Value = Pretax\ Value - Taxes\ Due\]

Subtract the expected tax bill to estimate after-tax proceeds.

0996 / Personal Finance and Tax Planning

Tax-adjusted gain

\[After\text{-}Tax\ Gain = Gain\times(1-Tax\ Rate)\]

Taxes reduce the gain actually kept by the investor.

0997 / Personal Finance and Tax Planning

Capital gains tax

\[Capital\ Gains\ Tax = Gain \times Capital\ Gains\ Rate\]

Capital-gains tax applies to the taxable investment profit.

0998 / Personal Finance and Tax Planning

Tax-loss harvest offset

\[Net\ Gain = Gains - Losses\]

Realized losses can offset realized gains.

0999 / Personal Finance and Tax Planning

Required gross return for target net return

\[Gross\ Return = \frac{Net\ Return}{1-Tax\ Rate}\]

Invert the after-tax return formula to solve for the required pretax return.

1000 / Personal Finance and Tax Planning

Retirement savings from employer match

\[Total\ Contribution = Employee\ Contribution + Employer\ Match\]

Employer matching increases total retirement-plan savings.