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Browse essential math and science formulas, from algebra, geometry, and trigonometry to physics, chemistry, and biology. Search by name or by variable, open any formula for a plain-language explanation and a worked example, then try the built-in calculator and jump to related practice.

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145 formulas

algebra

Quadratic Formula

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Finds the roots of any quadratic equation ax² + bx + c = 0.

algebra

Slope-Intercept Form

y=mx+by = mx + b

The equation of a line where m is the slope and b is the y-intercept.

algebra

Percent of a Number

P=r100⋅QP = \frac{r}{100} \cdot Q

Finds r% of a whole quantity Q — the part P. Also used for discounts, tax, and tips when r is the percent rate.

algebra

Point-Slope Form

y−y1=m(x−x1)y - y_1 = m(x - x_1)

Equation of a line using a known point (x₁, y₁) and the slope m.

algebra

Distance Formula

d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Calculates the straight-line distance between two points on a coordinate plane.

algebra

Midpoint Formula

M=(x1+x22, y1+y22)M = \left(\frac{x_1 + x_2}{2},\, \frac{y_1 + y_2}{2}\right)

Finds the exact middle point between two coordinates.

algebra

Slope Formula

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

Calculates the slope (steepness) of a line between two points.

algebra

Rational Number Form

r=ab,  a,b∈Z,  b≠0r = \frac{a}{b},\; a,b \in \mathbb{Z},\; b \neq 0

A rational number can be written as a fraction of two integers with a nonzero denominator.

algebra

Real Number Hierarchy

N⊂Z⊂Q⊂R\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}

Natural numbers are inside integers, integers are inside rational numbers, and rational plus irrational numbers make all real numbers.

algebra

Square Root Bounds

n2<m<(n+1)2⇒n<m<n+1n^2 < m < (n+1)^2 \Rightarrow n < \sqrt{m} < n+1

To estimate a nonperfect square root, find the two consecutive perfect squares around the number.

geometry

Area of a Circle

A=πr2A = \pi r^2

Calculates the area enclosed by a circle with radius r.

geometry

Circumference of a Circle

C=2πrC = 2\pi r

Calculates the perimeter (distance around) a circle.

geometry

Area of a Triangle

A=12bhA = \frac{1}{2}bh

Calculates the area of a triangle given its base and height.

geometry

Pythagorean Theorem

a2+b2=c2a^2 + b^2 = c^2

In a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. This is one of the most important formulas in all of mathematics.

geometry

Volume of a Cylinder

V=πr2hV = \pi r^2 h

Calculates the volume of a cylinder given its radius and height.

trigonometry

Sine (SOH)

sin⁡θ=oppositehypotenuse\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}

In a right triangle, sine of an angle equals the opposite side divided by the hypotenuse.

trigonometry

Cosine (CAH)

cos⁡θ=adjacenthypotenuse\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}

In a right triangle, cosine of an angle equals the adjacent side divided by the hypotenuse.

trigonometry

Tangent (TOA)

tan⁡θ=oppositeadjacent\tan\theta = \frac{\text{opposite}}{\text{adjacent}}

In a right triangle, tangent of an angle equals the opposite side divided by the adjacent side.

trigonometry

Pythagorean Identity

sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1

A fundamental trigonometric identity that holds true for all angles θ.

physics

Newton's Second Law

F=maF = ma

Force equals mass times acceleration — the foundation of classical mechanics.

physics

Velocity-Time Equation

v=v0+atv = v_0 + at

Calculates final velocity given initial velocity, acceleration, and time.

physics

Displacement Equation

d=v0t+12at2d = v_0 t + \frac{1}{2}at^2

Calculates displacement given initial velocity, acceleration, and time.

physics

Ohm's Law

V=IRV = IR

Voltage equals current times resistance in an electrical circuit.

physics

Work Formula

W=Fdcos⁡θW = Fd\cos\theta

Work equals force times displacement times the cosine of the angle between them.

chemistry

Ideal Gas Law

PV=nRTPV = nRT

Relates pressure, volume, amount, and temperature of an ideal gas.

chemistry

Density Formula

ρ=mV\rho = \frac{m}{V}

Density is mass per unit volume: how much matter is packed into a space.

chemistry

Mole Conversion

n=mMn = \frac{m}{M}

Converts between grams and moles using molar mass.

biology

Hardy-Weinberg Equation

p2+2pq+q2=1p^2 + 2pq + q^2 = 1

Predicts genotype frequencies in a non-evolving population. p and q are allele frequencies.

biology

Population Growth Rate

dNdt=rN\frac{dN}{dt} = rN

Exponential growth model where the rate of population change depends on the growth rate and current population size.

algebra

Vertex Formula

x=−b2ax = \frac{-b}{2a}

Finds the x-coordinate of the vertex of a parabola ax² + bx + c.

algebra

Discriminant

Δ=b2−4ac\Delta = b^2 - 4ac

Determines the nature of roots in a quadratic equation.

algebra

Standard Form of a Line

Ax+By=CAx + By = C

General form of a linear equation.

algebra

Difference of Squares

a2−b2=(a+b)(a−b)a^{2} - b^{2} = (a + b)(a - b)

Factorization pattern for difference of two perfect squares.

algebra

Perfect Square Trinomial

a2±2ab+b2=(a±b)2a^2 \pm 2ab + b^2 = (a \pm b)^2

Factorization pattern for perfect square trinomials.

algebra

Product Rule for Exponents

am⋅an=am+na^m \cdot a^n = a^{m+n}

When multiplying with same base, add the exponents.

algebra

Quotient Rule for Exponents

aman=am−n\frac{a^m}{a^n} = a^{m-n}

When dividing with same base, subtract the exponents.

algebra

Power Rule for Exponents

(am)n=amn(a^m)^n = a^{mn}

When raising a power to a power, multiply the exponents.

algebra

Scientific Notation

N=a×10n(1≤a<10)N = a \times 10^n \quad (1 \le a < 10)

Write very large or small magnitudes as a coefficient times a power of ten.

algebra

Logarithm Definition

log⁡ba=c  ⟺  bc=a\log_b a = c \iff b^c = a

Logarithm is the inverse of exponentiation.

algebra

Logarithm Product Rule

log⁡b(ab)=log⁡ba+log⁡bb\log_b(ab) = \log_b a + \log_b b

Log of a product equals the sum of the logs.

algebra

Logarithm Quotient Rule

log⁡b(ab)=log⁡ba−log⁡bb\log_b(\frac{a}{b}) = \log_b a - \log_b b

Log of a quotient equals the difference of the logs.

algebra

Polynomial Remainder Theorem

P(x)=(x−c)Q(x)+P(c)P(x) = (x - c)Q(x) + P(c)

When a polynomial P(x) is divided by (x − c), the remainder is the constant P(c) — the value of P at x = c.

algebra

Polynomial Factor Theorem

(x−c) is a factor of P(x)  ⟺  P(c)=0(x - c) \text{ is a factor of } P(x) \iff P(c) = 0

A linear factor (x − c) divides P(x) exactly when c is a root: substituting x = c gives zero.

algebra

Rational Root Theorem (statement)

pq (lowest terms)  ⟹  p∣a0,q∣an\dfrac{p}{q}\ \text{(lowest terms)} \implies p \mid a_{0},\quad q \mid a_{n}

Every possible rational root p/q (in lowest terms) must have p dividing the constant term and q dividing the leading coefficient — a finite checklist of candidates.

algebra

Vieta's Formulas (Quadratic)

r1+r2=−ba,r1r2=car_1 + r_2 = -\frac{b}{a}, \quad r_1 r_2 = \frac{c}{a}

For ax² + bx + c = 0 with roots r₁ and r₂, the sum of roots is −b/a and the product is c/a.

algebra

Partial Fractions (Distinct Linear Factors)

P(x)(x−r1)(x−r2)=Ax−r1+Bx−r2(r1≠r2)\frac{P(x)}{(x - r_1)(x - r_2)} = \frac{A}{x - r_1} + \frac{B}{x - r_2} \quad (r_1 \neq r_2)

A proper rational function with distinct linear factors in the denominator splits into one simple term per factor; solve for A and B by clearing denominators or substituting convenient x values.

algebra

Absolute Value Inequality |x| < a

∣x∣<a  ⟺  −a<x<a(a>0)|x| < a \iff -a < x < a \quad (a > 0)

For positive a, |x| < a means x is strictly between −a and a — a bounded interval.

algebra

Absolute Value Inequality |x| > a

∣x∣>a  ⟺  x<−a   or   x>a(a>0)|x| > a \iff x < {-a} \;\text{ or }\; x > a \quad (a > 0)

For positive a, |x| > a means x is farther than a from 0 — two rays, not a single interval.

algebra

Sign Chart for Polynomial / Rational Inequalities

Factor f  ⇒  zeros/poles on a line   ⇒  sign (+) / (−) per interval\text{Factor } f \;\Rightarrow\; \text{zeros/poles on a line } \;\Rightarrow\; \text{sign } (+)\,/\,(-)\ \text{per interval}

Factor the expression, plot all roots and vertical asymptotes on a number line, pick a test point in each interval, and read where the product/quotient is positive or negative. Signs often flip at simple zeros; poles make intervals where f is undefined.

algebra

Cramer's Rule (2×2)

x=∣ebfd∣∣abcd∣,y=∣aecf∣∣abcd∣x = \frac{\begin{vmatrix} e & b \\ f & d \end{vmatrix}}{\begin{vmatrix} a & b \\ c & d \end{vmatrix}}, \quad y = \frac{\begin{vmatrix} a & e \\ c & f \end{vmatrix}}{\begin{vmatrix} a & b \\ c & d \end{vmatrix}}

For ax + by = e and cx + dy = f, replace the column of x or y coefficients with the constants (e, f) in the numerator determinant; denominator is the coefficient determinant (must be nonzero).

algebra

Determinant of a 2×2 Matrix

det⁡[abcd]=ad−bc\det\begin{bmatrix} a & b \\ c & d \end{bmatrix} = ad - bc

The determinant of [[a,b],[c,d]] is ad − bc; zero means the rows are linearly dependent.

algebra

Inverse of a 2×2 Matrix

[abcd]−1=1ad−bc[d−b−ca]\begin{bmatrix} a & b \\ c & d \end{bmatrix}^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}

Swap the diagonal entries, negate the off-diagonals, and divide by the determinant; exists only when ad − bc ≠ 0.

algebra

Rational Exponent as a Radical

am/n=amn=(an)m(a>0 typical)a^{m/n} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m \quad (a > 0 \text{ typical})

A rational exponent m/n means take the nth root, then raise to the mth power (or the equivalent order when defined).

algebra

Rationalizing a Monomial Radical Denominator

1a⋅aa=aa(a>0)\frac{1}{\sqrt{a}} \cdot \frac{\sqrt{a}}{\sqrt{a}} = \frac{\sqrt{a}}{a} \quad (a > 0)

Multiply numerator and denominator by √a so the denominator becomes the rational number a.

algebra

Change-of-Base Formula

log⁡bx=log⁡cxlog⁡cb(b,x,c>0,  b≠1,  c≠1)\log_b x = \frac{\log_c x}{\log_c b} \quad (b, x, c > 0,\; b \neq 1,\; c \neq 1)

Rewrite any base-b logarithm using a calculator-friendly base c (often 10 or e).

algebra

Logarithm Power Rule

log⁡b(xp)=plog⁡bx(x>0,  b>0,  b≠1)\log_b(x^p) = p\log_b x \quad (x > 0,\; b > 0,\; b \neq 1)

An exponent inside the log moves out as a multiplier — the log version of a power law.

algebra

Continuous Compound Growth

A=PertA = Pe^{rt}

When interest (or growth) compounds continuously at rate r per unit time, amount A grows exponentially from principal P — distinct from discrete compounding or half-life decay models.

geometry

Circle (Standard Form)

(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2

A circle with center (h, k) and radius r: all points at fixed distance r from the center.

geometry

Parabola Opening Up/Down (Standard)

(x−h)2=4p(y−k)(x - h)^2 = 4p(y - k)

Vertical-axis parabola with vertex (h, k): |p| is focal distance from vertex; p>0 opens up, p<0 opens down — complements vertex x = −b/(2a) for quadratics in x.

algebra

Arithmetic Sequence (nth Term)

an=a1+(n−1)da_n = a_1 + (n - 1)d

Each term adds a common difference d to the previous one; aₙ is the value at position n.

algebra

Arithmetic Series (Partial Sum)

Sn=n2(a1+an)S_n = \frac{n}{2}(a_1 + a_n)

Sum of the first n terms of an arithmetic sequence — average of first and last times how many terms.

algebra

Geometric Sequence (nth Term)

an=a1rn−1(r≠0 typical)a_n = a_1 r^{n - 1} \quad (r \neq 0 \text{ typical})

Each term multiplies the previous by common ratio r; growth or decay by a constant factor.

algebra

Finite Geometric Series Sum

Sn=a11−rn1−r(r≠1)S_n = a_1 \frac{1 - r^n}{1 - r} \quad (r \neq 1)

Sum of the first n terms of a geometric sequence; for r = 1 use Sₙ = n·a₁ instead.

algebra

Binomial Theorem

(a+b)n=∑k=0n(nk)an−kbk(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

Expanding (a + b)ⁿ: coefficients are binomial coefficients; each term has total degree n.

geometry

Area of a Rectangle

A=lwA = lw

Calculates the area of a rectangle given length and width.

geometry

Perimeter of a Rectangle

P=2(l+w)P = 2(l + w)

Calculates the perimeter of a rectangle.

geometry

Area of a Square

A=s2A = s^2

Calculates the area of a square given side length.

geometry

Perimeter of a Square

P=4sP = 4s

Calculates the perimeter of a square.

geometry

Area of a Trapezoid

A=12(b1+b2)hA = \frac{1}{2}(b_1 + b_2)h

Calculates the area of a trapezoid given parallel sides and height.

geometry

Area of a Parallelogram

A=bhA = bh

Calculates the area of a parallelogram given base and height.

geometry

Surface Area of a Sphere

A=4πr2A = 4\pi r^2

Calculates the surface area of a sphere.

geometry

Volume of a Sphere

V=43πr3V = \frac{4}{3}\pi r^3

Calculates the volume of a sphere.

geometry

Volume of a Cone

V=13πr2hV = \frac{1}{3}\pi r^2 h

Calculates the volume of a cone.

geometry

Surface Area of a Cone

A=πr(r+l)A = \pi r(r + l)

Calculates the total surface area of a cone.

geometry

Volume of a Cube

V=s3V = s^3

Calculates the volume of a cube given its side length.

geometry

Volume of a Rectangular Prism

V=lwhV = lwh

Calculates the volume of a rectangular prism (box).

geometry

Surface Area of a Rectangular Prism

A=2(lw+lh+wh)A = 2(lw + lh + wh)

Calculates the total surface area of a rectangular prism.

geometry

Volume of a Pyramid

V=13BhV = \frac{1}{3}Bh

Calculates the volume of a pyramid.

trigonometry

Law of Sines

asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Relates side lengths to sines of opposite angles in any triangle.

trigonometry

Law of Cosines

c2=a2+b2−2abcos⁡Cc^2 = a^2 + b^2 - 2ab\cos C

Generalization of Pythagorean theorem for any triangle.

trigonometry

Double Angle: Sine

sin⁡(2θ)=2sin⁡θcos⁡θ\sin(2\theta) = 2\sin\theta\cos\theta

Double angle identity for sine.

trigonometry

Double Angle: Cosine

cos⁡(2θ)=cos⁡2θ−sin⁡2θ\cos(2\theta) = \cos^2\theta - \sin^2\theta

Double angle identity for cosine.

trigonometry

Sum Angle: Sine

sin⁡(α+β)=sin⁡αcos⁡β+cos⁡αsin⁡β\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta

Sum angle identity for sine.

trigonometry

Sum Angle: Cosine

cos⁡(α+β)=cos⁡αcos⁡β−sin⁡αsin⁡β\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta

Sum angle identity for cosine.

physics

Kinetic Energy

KE=12mv2KE = \frac{1}{2}mv^2

Energy of motion depends on mass and velocity squared.

physics

Gravitational Potential Energy

PE=mghPE = mgh

Energy stored due to height in a gravitational field.

physics

Momentum

p=mvp = mv

Product of mass and velocity.

physics

Impulse

J=FΔt=ΔpJ = F\Delta t = \Delta p

Change in momentum equals impulse.

physics

Power

P=Wt=FvP = \frac{W}{t} = Fv

Rate of doing work or energy transfer.

physics

Frequency-Period Relationship

f=1Tf = \frac{1}{T}

Frequency is the reciprocal of period.

physics

Wave Speed

v=fλv = f\lambda

Wave speed equals frequency times wavelength.

physics

Newton's Law of Universal Gravitation

F=Gm1m2r2F = G\frac{m_1 m_2}{r^2}

Gravitational force between two masses.

physics

Centripetal Force

Fc=mv2rF_c = \frac{mv^2}{r}

Force required to keep an object moving in a circle.

physics

Centripetal Acceleration

ac=v2ra_c = \frac{v^2}{r}

Acceleration toward the center of circular motion.

chemistry

Molar Concentration

C=nVC = \frac{n}{V}

Moles of solute per liter of solution.

chemistry

Dilution Formula

M1V1=M2V2M_1V_1 = M_2V_2

Relationship between concentration and volume before and after dilution.

chemistry

Percent Composition

%=melementmcompound×100\% = \frac{m_{element}}{m_{compound}} \times 100

Mass percentage of an element in a compound.

chemistry

Empirical Formula

Ratio=molesGCD\text{Ratio} = \frac{\text{moles}}{\text{GCD}}

Simplest whole number ratio of elements in a compound.

chemistry

pH Calculation

pH=−log⁡[H+]\text{pH} = -\log[H^+]

Measures acidity or basicity of a solution.

chemistry

pOH Calculation

pOH=−log⁡[OH−]\text{pOH} = -\log[OH^-]

Measures hydroxide ion concentration.

chemistry

pH-pOH Relationship

pH+pOH=14\text{pH} + \text{pOH} = 14

Sum of pH and pOH always equals 14 at 25°C.

chemistry

Gibbs Free Energy

ΔG=ΔH−TΔS\Delta G = \Delta H - T\Delta S

Determines spontaneity of a reaction.

biology

Logistic Growth

dNdt=rN(1−NK)\frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right)

Population growth limited by carrying capacity.

biology

Half-Life

N=N0(12)t/t1/2N = N_0(\frac{1}{2})^{t/t_{1/2}}

Time for half of a substance to decay or be eliminated.

biology

Exponential Decay

N=N0e−ktN = N_0 e^{-kt}

General exponential decay model.

biology

Shannon Diversity Index

H′=−∑i=1spiln⁡(pi)H' = -\sum_{i=1}^{s} p_i \ln(p_i)

Measures species diversity in a community.

biology

Simpson Diversity Index

D=1−∑i=1spi2D = 1 - \sum_{i=1}^{s} p_i^2

Probability that two randomly selected individuals are different species.

biology

Michaelis-Menten Equation

v=Vmax[S]Km+[S]v = \frac{V_{max}[S]}{K_m + [S]}

Rate of enzymatic reactions.

biology

Lineweaver-Burk Plot

1v=KmVmax⋅1[S]+1Vmax\frac{1}{v} = \frac{K_m}{V_{max}} \cdot \frac{1}{[S]} + \frac{1}{V_{max}}

Linear transformation of Michaelis-Menten equation.

geometry

Triangle Angle Sum

A+B+C=180∘A + B + C = 180^\circ

The three interior angles of any triangle always add up to 180 degrees.

geometry

Polygon Interior Angle Sum

S=(n−2)⋅180∘S = (n - 2)\cdot 180^\circ

The interior angles of an n-sided polygon add up to (n − 2) × 180 degrees.

geometry

Regular Polygon Angles

Interior=(n−2)⋅180∘n,Exterior=360∘n\text{Interior} = \frac{(n - 2)\cdot 180^\circ}{n}, \quad \text{Exterior} = \frac{360^\circ}{n}

Each interior and exterior angle of a regular polygon (all sides and angles equal) with n sides.

geometry

Arc Length

s=θ360∘⋅2πrs = \frac{\theta}{360^\circ}\cdot 2\pi r

The length of a circular arc that spans a central angle θ (in degrees) in a circle of radius r.

geometry

Area of a Sector

A=θ360∘⋅πr2A = \frac{\theta}{360^\circ}\cdot \pi r^2

The area of a pie-slice sector that spans a central angle θ (in degrees) in a circle of radius r.

geometry

Surface Area of a Cylinder

SA=2πr2+2πrhSA = 2\pi r^2 + 2\pi r h

Total surface area of a closed cylinder: two circular ends plus the curved side.

geometry

Surface Area of a Cube

SA=6s2SA = 6 s^2

Total surface area of a cube with edge length s (six equal square faces).

geometry

Surface Area of a Square Pyramid

SA=b2+2blSA = b^2 + 2 b l

Surface area of a pyramid with a square base of side b and slant height l: the base plus four triangular faces.

geometry

Area of a Rhombus or Kite

A=12d1d2A = \frac{1}{2} d_1 d_2

Area of a rhombus or kite equals half the product of its two diagonals d₁ and d₂.

geometry

Area of a Regular Polygon

A=12aPA = \frac{1}{2} a P

Area of a regular polygon equals half the apothem a times the perimeter P.

geometry

Heron's Formula

A=s(s−a)(s−b)(s−c),s=a+b+c2A = \sqrt{s(s-a)(s-b)(s-c)}, \quad s = \frac{a+b+c}{2}

Finds a triangle's area from its three side lengths a, b, c, using the semi-perimeter s.

geometry

Special Right Triangles

45-45-90: 1:1:230-60-90: 1:3:245\text{-}45\text{-}90:\ 1 : 1 : \sqrt{2} \qquad 30\text{-}60\text{-}90:\ 1 : \sqrt{3} : 2

Fixed side ratios for the two special right triangles, so you can find sides without trigonometry.

geometry

Triangle Inequality

a+b>c,a+c>b,b+c>aa + b > c, \quad a + c > b, \quad b + c > a

Three lengths form a triangle only if each side is shorter than the sum of the other two.

geometry

Similar Triangles (Proportional Sides)

aa′=bb′=cc′\frac{a}{a'} = \frac{b}{b'} = \frac{c}{c'}

Similar triangles have equal angles and proportional corresponding sides; set up a proportion to find a missing length.

algebra

Simple Interest

I=PrtI = P r t

Interest earned (or owed) on a principal P at annual rate r for t years. Enter r as a decimal (0.05 for 5%).

algebra

Compound Interest

A=P(1+rn)ntA = P\left(1 + \frac{r}{n}\right)^{n t}

Final balance when interest compounds n times per year. Enter r as a decimal (0.05 for 5%).

algebra

Exponential Growth & Decay

A=a(1+r)tA = a(1 + r)^t

A quantity that grows (r > 0) or decays (r < 0) by a fixed percent each period. Enter r as a decimal.

algebra

Distance = Rate × Time

d=rtd = r t

Distance traveled equals speed (rate) times time, the basis of motion word problems.

algebra

Direct Variation

y=kxy = k x

y varies directly with x: as x increases, y increases by a constant factor k.

algebra

Inverse Variation

y=kxy = \frac{k}{x}

y varies inversely with x: as x increases, y decreases so that their product stays constant.

algebra

Percent Change

percent change=V2−V1V1⋅100%\text{percent change} = \frac{V_2 - V_1}{V_1}\cdot 100\%

How much a value increased or decreased, as a percent of the original. Positive is an increase, negative a decrease.

algebra

Sum of Cubes

a3+b3=(a+b)(a2−ab+b2)a^3 + b^3 = (a + b)(a^2 - a b + b^2)

Factoring pattern for a sum of two perfect cubes.

algebra

Difference of Cubes

a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + a b + b^2)

Factoring pattern for a difference of two perfect cubes.

algebra

Parallel & Perpendicular Slopes

m∥=m,m⊥=−1mm_{\parallel} = m, \quad m_{\perp} = -\frac{1}{m}

Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals.

algebra

Modulus of a Complex Number

∣a+bi∣=a2+b2|a + bi| = \sqrt{a^2 + b^2}

The size (distance from the origin) of a complex number a + bi, where i is the imaginary unit with i² = −1.

algebra

Function Composition

(f∘g)(x)=f(g(x))(f \circ g)(x) = f\bigl(g(x)\bigr)

Applying one function to the output of another: do g first, then feed the result into f.

algebra

Inverse Function

f(f−1(x))=xandf−1(f(x))=xf\bigl(f^{-1}(x)\bigr) = x \quad\text{and}\quad f^{-1}\bigl(f(x)\bigr) = x

An inverse function undoes the original: swap x and y and solve for y. Each composition returns the input.

algebra

Infinite Geometric Series

S=a1−r,∣r∣<1S = \frac{a}{1 - r}, \quad |r| < 1

The sum of an infinite geometric series with first term a and common ratio r, valid only when |r| < 1.

algebra

Permutations

nPr=n!(n−r)!{}_nP_r = \frac{n!}{(n - r)!}

The number of ways to arrange r items from n distinct items when order matters.

algebra

Combinations

nCr=n!r! (n−r)!{}_nC_r = \frac{n!}{r!\,(n - r)!}

The number of ways to choose r items from n distinct items when order does NOT matter.

algebra

Basic Probability

P(E)=favorable outcomestotal outcomesP(E) = \frac{\text{favorable outcomes}}{\text{total outcomes}}

The probability of an event with equally likely outcomes: favorable outcomes divided by total outcomes.

geometry

Ellipse (Standard Form)

x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

An ellipse centered at the origin with horizontal semi-axis a and vertical semi-axis b.

geometry

Hyperbola (Standard Form)

x2a2−y2b2=1\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

A hyperbola centered at the origin opening left and right, with vertices at (±a, 0).

trigonometry

Reciprocal Trig Identities

csc⁡θ=1sin⁡θ,sec⁡θ=1cos⁡θ,cot⁡θ=1tan⁡θ\csc\theta = \frac{1}{\sin\theta}, \quad \sec\theta = \frac{1}{\cos\theta}, \quad \cot\theta = \frac{1}{\tan\theta}

Cosecant, secant, and cotangent are the reciprocals of sine, cosine, and tangent.

trigonometry

Radian and Degree Conversion

radians=degrees⋅π180∘\text{radians} = \text{degrees}\cdot \frac{\pi}{180^\circ}

Convert an angle from degrees to radians by multiplying by π/180 (reverse by 180/π).

trigonometry

Area of a Triangle (SAS)

A=12absin⁡CA = \frac{1}{2}ab\sin C

A triangle's area from two sides a, b and the angle C between them (the included angle, in degrees).

About the formula library

The formula library collects the equations students use across math and science, algebra, geometry, trigonometry, and calculus alongside physics, chemistry, and biology, each with a plain-language explanation of what it means and when to use it.

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