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Math Formula Encyclopedia

Search, understand, and use formulas across arithmetic, algebra, geometry, trigonometry, calculus, statistics, finance, science, engineering, and everyday math.

134 formulas10 subjectsExamples + variables
AllAlgebra 16Arithmetic 8Calculus 15Everyday Math & Conversions 7Finance & Business 9Geometry 28Number Theory & Sequences 8Science & Engineering 16Statistics & Probability 16Trigonometry 11

134 matching formulas

Arithmetic

Percentage of a Number

P = (r/100) × N

Find a percentage of a quantity.

Open formula →Arithmetic

Percentage Change

% change = ((new − old) / old) × 100

Measures relative increase or decrease from an original value.

Open formula →Arithmetic

Average (Arithmetic Mean)

x̄ = (x₁ + x₂ + … + xₙ) / n

Adds all values and divides by the count.

Open formula →Arithmetic

Weighted Average

x̄w = Σ(wᵢxᵢ) / Σwᵢ

Average in which values contribute according to assigned weights.

Open formula →Arithmetic

Ratio

a : b = a/b

Compares two quantities by division.

Open formula →Arithmetic

Proportion Cross Product

a/b = c/d ⇒ ad = bc

Solves equivalent ratios by equating cross products.

Open formula →Arithmetic

Simple Interest

I = Prt

Calculates interest without compounding.

Open formula →Arithmetic

Compound Amount

A = P(1 + r/n)^(nt)

Future value with periodic compounding.

Open formula →Algebra

Linear Equation

x = (c − b) / a

Solves ax + b = c for x.

Open formula →Algebra

Slope

m = (y₂ − y₁)/(x₂ − x₁)

Rate of vertical change per unit horizontal change.

Open formula →Algebra

Slope-Intercept Form

y = mx + b

Standard equation of a nonvertical straight line.

Open formula →Algebra

Point-Slope Form

y − y₁ = m(x − x₁)

Equation of a line from a point and slope.

Open formula →Algebra

Quadratic Formula

x = (−b ± √(b² − 4ac))/(2a)

Finds roots of a quadratic equation.

Open formula →Algebra

Discriminant

Δ = b² − 4ac

Determines the nature of quadratic roots.

Open formula →Algebra

Vertex of a Parabola

xᵥ = −b/(2a)

Finds the horizontal coordinate of a quadratic vertex.

Open formula →Algebra

Difference of Squares

a² − b² = (a−b)(a+b)

Factors a difference of perfect squares.

Open formula →Algebra

Exponent Product Rule

a^m · a^n = a^(m+n)

Adds exponents when multiplying like bases.

Open formula →Algebra

Exponent Quotient Rule

a^m / a^n = a^(m−n)

Subtracts exponents when dividing like bases.

Open formula →Algebra

Power of a Power

(a^m)^n = a^(mn)

Multiplies exponents when raising a power to a power.

Open formula →Algebra

Logarithm Change of Base

log_b(x)=ln(x)/ln(b)

Rewrites a logarithm using another base.

Open formula →Geometry

Rectangle Area

A = lw

Area enclosed by a rectangle.

Open formula →Geometry

Rectangle Perimeter

P = 2(l+w)

Distance around a rectangle.

Open formula →Geometry

Square Area

A = s²

Area of a square.

Open formula →Geometry

Square Perimeter

P = 4s

Distance around a square.

Open formula →Geometry

Triangle Area

A = ½bh

Area of a triangle.

Open formula →Geometry

Heron’s Formula

A = √(s(s−a)(s−b)(s−c))

Area of a triangle from three side lengths.

Open formula →Geometry

Pythagorean Theorem

c² = a² + b²

Relates side lengths of a right triangle.

Open formula →Geometry

Circle Circumference

C = 2πr

Distance around a circle.

Open formula →Geometry

Circle Area

A = πr²

Area enclosed by a circle.

Open formula →Geometry

Arc Length

L = rθ

Length of a circular arc.

Open formula →Geometry

Sector Area

A = ½r²θ

Area of a circular sector.

Open formula →Geometry

Trapezoid Area

A = ½(a+b)h

Area of a trapezoid.

Open formula →Geometry

Parallelogram Area

A = bh

Area of a parallelogram.

Open formula →Geometry

Rhombus Area

A = ½d₁d₂

Area of a rhombus from diagonals.

Open formula →Geometry

Ellipse Area

A = πab

Area enclosed by an ellipse.

Open formula →Geometry

Cube Volume

V = s³

Volume of a cube.

Open formula →Geometry

Rectangular Prism Volume

V = lwh

Volume of a rectangular prism.

Open formula →Geometry

Cylinder Volume

V = πr²h

Volume of a right circular cylinder.

Open formula →Geometry

Cone Volume

V = (1/3)πr²h

Volume of a cone.

Open formula →Geometry

Sphere Volume

V = (4/3)πr³

Volume of a sphere.

Open formula →Geometry

Sphere Surface Area

S = 4πr²

Surface area of a sphere.

Open formula →Geometry

Distance Between Two Points

d = √((x₂−x₁)²+(y₂−y₁)²)

Euclidean distance in a plane.

Open formula →Geometry

Midpoint

M = ((x₁+x₂)/2,(y₁+y₂)/2)

Coordinates halfway between two points.

Open formula →Trigonometry

Sine Ratio

sin θ = opposite/hypotenuse

Right-triangle sine ratio.

Open formula →Trigonometry

Cosine Ratio

cos θ = adjacent/hypotenuse

Right-triangle cosine ratio.

Open formula →Trigonometry

Tangent Ratio

tan θ = opposite/adjacent

Right-triangle tangent ratio.

Open formula →Trigonometry

Pythagorean Identity

sin²θ + cos²θ = 1

Fundamental identity connecting sine and cosine.

Open formula →Trigonometry

Law of Sines

a/sin A = b/sin B = c/sin C

Relates sides and opposite angles in any triangle.

Open formula →Trigonometry

Law of Cosines

c² = a² + b² − 2ab cos C

Generalizes the Pythagorean theorem to any triangle.

Open formula →Trigonometry

Degrees to Radians

radians = degrees × π/180

Converts degrees to radians.

Open formula →Trigonometry

Radians to Degrees

degrees = radians × 180/π

Converts radians to degrees.

Open formula →Calculus

Derivative Definition

f′(x)=lim(h→0)[f(x+h)−f(x)]/h

Defines instantaneous rate of change.

Open formula →Calculus

Power Rule

d/dx(x^n)=n x^(n−1)

Derivative of a power function.

Open formula →Calculus

Product Rule

(fg)′ = f′g + fg′

Derivative of a product.

Open formula →Calculus

Quotient Rule

(f/g)′=(f′g−fg′)/g²

Derivative of a quotient.

Open formula →Calculus

Chain Rule

d/dx f(g(x)) = f′(g(x))g′(x)

Derivative of a composite function.

Open formula →Calculus

Integral Power Rule

∫x^n dx = x^(n+1)/(n+1)+C

Antiderivative of a power.

Open formula →Calculus

Fundamental Theorem of Calculus

∫ₐᵇ f(x)dx = F(b)−F(a)

Connects definite integrals with antiderivatives.

Open formula →Calculus

Average Rate of Change

[f(b)−f(a)]/(b−a)

Slope of a secant line over an interval.

Open formula →Calculus

Average Value of a Function

f_avg=(1/(b−a))∫ₐᵇf(x)dx

Mean value of a continuous function over an interval.

Open formula →Statistics & Probability

Population Mean

μ = Σx / N

Arithmetic mean of an entire population.

Open formula →Statistics & Probability

Sample Mean

x̄ = Σx / n

Arithmetic mean of a sample.

Open formula →Statistics & Probability

Population Variance

σ² = Σ(x−μ)²/N

Average squared deviation from the population mean.

Open formula →Statistics & Probability

Sample Variance

s² = Σ(x−x̄)²/(n−1)

Unbiased estimator of population variance.

Open formula →Statistics & Probability

Standard Deviation

σ = √σ²

Square root of variance.

Open formula →Statistics & Probability

Z-Score

z = (x−μ)/σ

Standardizes a value by distance from the mean in standard deviations.

Open formula →Statistics & Probability

Combination

C(n,r)=n!/[r!(n−r)!]

Counts unordered selections without replacement.

Open formula →Statistics & Probability

Permutation

P(n,r)=n!/(n−r)!

Counts ordered selections without replacement.

Open formula →Statistics & Probability

Basic Probability

P(E)=favorable outcomes/total outcomes

Probability for equally likely outcomes.

Open formula →Statistics & Probability

Complement Rule

P(Eᶜ)=1−P(E)

Probability that an event does not occur.

Open formula →Statistics & Probability

Addition Rule

P(A∪B)=P(A)+P(B)−P(A∩B)

Probability that A or B occurs.

Open formula →Statistics & Probability

Multiplication Rule

P(A∩B)=P(A)P(B|A)

Probability that both events occur.

Open formula →Statistics & Probability

Binomial Probability

P(X=k)=C(n,k)p^k(1−p)^(n−k)

Probability of exactly k successes in independent Bernoulli trials.

Open formula →Statistics & Probability

Expected Value

E[X]=ΣxP(x)

Long-run probability-weighted average outcome.

Open formula →Finance & Business

Future Value

FV = PV(1+r)^n

Future value under periodic compounding.

Open formula →Finance & Business

Present Value

PV = FV/(1+r)^n

Present value of a future amount.

Open formula →Finance & Business

Loan Payment

PMT = P·[r(1+r)^n]/[(1+r)^n−1]

Fixed payment for a fully amortizing loan.

Open formula →Finance & Business

Debt-to-Income Ratio

DTI = monthly debt / gross monthly income × 100

Common affordability ratio.

Open formula →Finance & Business

Markup Percentage

markup% = (selling price−cost)/cost ×100

Markup measured relative to cost.

Open formula →Finance & Business

Profit Margin

margin% = profit/revenue ×100

Profit measured relative to revenue.

Open formula →Finance & Business

Break-Even Units

Q = fixed costs/(price−variable cost per unit)

Units required to cover fixed and variable costs.

Open formula →Finance & Business

CAGR

CAGR=(ending/beginning)^(1/n)−1

Smoothed annualized growth rate.

Open formula →Science & Engineering

Speed

v = d/t

Average speed equals distance divided by time.

Open formula →Science & Engineering

Acceleration

a = Δv/Δt

Rate of change of velocity.

Open formula →Science & Engineering

Newton’s Second Law

F = ma

Net force equals mass times acceleration.

Open formula →Science & Engineering

Weight

W = mg

Gravitational force near a body surface.

Open formula →Science & Engineering

Momentum

p = mv

Linear momentum of a body.

Open formula →Science & Engineering

Kinetic Energy

KE = ½mv²

Energy of motion.

Open formula →Science & Engineering

Gravitational Potential Energy

PE = mgh

Near-surface gravitational potential energy.

Open formula →Science & Engineering

Work

W = Fd cos θ

Energy transferred by a force through displacement.

Open formula →Science & Engineering

Power

P = W/t

Rate of doing work.

Open formula →Science & Engineering

Ohm’s Law

V = IR

Relationship among voltage, current, and resistance.

Open formula →Science & Engineering

Electrical Power

P = VI

Electrical power in a circuit.

Open formula →Science & Engineering

Density

ρ = m/V

Mass per unit volume.

Open formula →Science & Engineering

Pressure

P = F/A

Force distributed over area.

Open formula →Number Theory & Sequences

Arithmetic Sequence Term

aₙ = a₁ + (n−1)d

Nth term of an arithmetic sequence.

Open formula →Number Theory & Sequences

Arithmetic Series Sum

Sₙ = n(a₁+aₙ)/2

Sum of an arithmetic series.

Open formula →Number Theory & Sequences

Geometric Sequence Term

aₙ = a₁r^(n−1)

Nth term of a geometric sequence.

Open formula →Number Theory & Sequences

Finite Geometric Series

Sₙ = a₁(1−r^n)/(1−r)

Sum of first n terms of a geometric series.

Open formula →Number Theory & Sequences

Infinite Geometric Series

S = a₁/(1−r)

Sum of a convergent infinite geometric series.

Open formula →Number Theory & Sequences

Euclidean Algorithm

gcd(a,b)=gcd(b,a mod b)

Repeatedly reduces a greatest-common-divisor problem.

Open formula →Number Theory & Sequences

Euler Totient for Prime

φ(p)=p−1

Counts positive integers up to p that are coprime to p.

Open formula →Number Theory & Sequences

Modular Congruence

a ≡ b (mod n) ⇔ n | (a−b)

Defines equivalence modulo n.

Open formula →Everyday Math & Conversions

Celsius to Fahrenheit

°F = (°C × 9/5) + 32

Temperature conversion.

Open formula →Everyday Math & Conversions

Fahrenheit to Celsius

°C = (°F − 32) × 5/9

Temperature conversion.

Open formula →Everyday Math & Conversions

Molarity

M = moles of solute / liters of solution

Concentration in moles per liter.

Open formula →Everyday Math & Conversions

Dilution

M₁V₁ = M₂V₂

Conservation relation for ideal dilution.

Open formula →Everyday Math & Conversions

BMI

BMI = mass(kg)/height(m)²

Body mass index calculation; it is a screening measure, not a diagnosis.

Open formula →Everyday Math & Conversions

Fuel Economy

MPG = miles driven / gallons used

Distance traveled per gallon.

Open formula →Everyday Math & Conversions

Unit Price

unit price = total price / quantity

Cost per unit for comparison shopping.

Open formula →Algebra

Binomial Square

(a+b)² = a²+2ab+b²

Common reference identity or relationship used in algebra.

Open formula →Algebra

Binomial Difference Square

(a−b)² = a²−2ab+b²

Common reference identity or relationship used in algebra.

Open formula →Algebra

Sum of Cubes

a³+b³=(a+b)(a²−ab+b²)

Common reference identity or relationship used in algebra.

Open formula →Algebra

Difference of Cubes

a³−b³=(a−b)(a²+ab+b²)

Common reference identity or relationship used in algebra.

Open formula →Geometry

Regular Polygon Interior Angle Sum

S=(n−2)×180°

Common reference identity or relationship used in geometry.

Open formula →Geometry

Regular Polygon Interior Angle

A=((n−2)×180°)/n

Common reference identity or relationship used in geometry.

Open formula →Geometry

Regular Polygon Exterior Angle

E=360°/n

Common reference identity or relationship used in geometry.

Open formula →Geometry

Prism Volume

V=Bh

Common reference identity or relationship used in geometry.

Open formula →Geometry

Pyramid Volume

V=(1/3)Bh

Common reference identity or relationship used in geometry.

Open formula →Trigonometry

Double-Angle Sine

sin(2θ)=2sinθ cosθ

Common reference identity or relationship used in trigonometry.

Open formula →Trigonometry

Double-Angle Cosine

cos(2θ)=cos²θ−sin²θ

Common reference identity or relationship used in trigonometry.

Open formula →Trigonometry

Tangent Identity

tanθ=sinθ/cosθ

Common reference identity or relationship used in trigonometry.

Open formula →Calculus

Derivative of e^x

d/dx(e^x)=e^x

Common reference identity or relationship used in calculus.

Open formula →Calculus

Derivative of ln x

d/dx(ln x)=1/x

Common reference identity or relationship used in calculus.

Open formula →Calculus

Derivative of sin x

d/dx(sin x)=cos x

Common reference identity or relationship used in calculus.

Open formula →Calculus

Derivative of cos x

d/dx(cos x)=−sin x

Common reference identity or relationship used in calculus.

Open formula →Calculus

Integral of 1/x

∫(1/x)dx=ln|x|+C

Common reference identity or relationship used in calculus.

Open formula →Calculus

Integral of e^x

∫e^x dx=e^x+C

Common reference identity or relationship used in calculus.

Open formula →Statistics & Probability

Standard Error of Mean

SE=σ/√n

Common reference identity or relationship used in statistics & probability.

Open formula →Statistics & Probability

Correlation Coefficient

r=Σ[(x−x̄)(y−ȳ)]/√(Σ(x−x̄)²Σ(y−ȳ)²)

Common reference identity or relationship used in statistics & probability.

Open formula →Finance & Business

Rule of 72

doubling years ≈ 72 / annual rate (%)

Common reference identity or relationship used in finance & business.

Open formula →Science & Engineering

Ideal Gas Law

PV=nRT

Common reference identity or relationship used in science & engineering.

Open formula →Science & Engineering

Wave Speed

v=fλ

Common reference identity or relationship used in science & engineering.

Open formula →Science & Engineering

Frequency and Period

f=1/T

Common reference identity or relationship used in science & engineering.

Open formula →
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