All formulas, A–Z

A

• Addition rule (or)$P(A\cup B)=P(A)+P(B)-P(A\cap B)$Probability and statistics
• Angle sum and difference formulas$\sin (A\pm B)=\sin A\cos B\pm \cos A\sin B,\cos (A\pm B)=\cos A\cos B\mp \sin A\sin B$Trigonometry
• Arc length$s=r\theta (\theta \text{in radians}),s=\frac{\theta }{360}\times 2\pi r(\theta \text{in degrees})$Geometry
• Area of a circle$A=\pi r^2$Geometry
• Area of a parallelogram$A=bh$Geometry
• Area of a rectangle$A=l\times w$Geometry
• Area of a regular polygon$A=\frac{n{s}^2}{4\tan \left(\frac{\pi }{n}\right)}$Geometry
• Area of a sector$A=\frac{1}{2}{r}^2\theta (\theta \text{in radians}),A=\frac{\theta }{360}\pi r^2(\theta \text{in degrees})$Geometry
• Area of a trapezoid (trapezium)$A=\frac{a+b}{2}h$Geometry
• Area of a triangle$A=\frac{1}{2}bh$Geometry
• Area of a triangle from two sides and an angle$A=\frac{1}{2}ab\sin C$Trigonometry
• Area of an ellipse$A=\pi ab$Geometry
• Arithmetic sequence (n-th term)$a_n=a_1+(n-1)d$Sequences and series
• Arithmetic series (sum)$S_n=\frac{n}{2}\left(2a_1+(n-1)d\right)$Sequences and series
• Average value of a function$\overline{f}=\frac{1}{b-a}{\int }_a^{b}f(x)dx$Calculus basics

B

• Bayes' theorem$P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}$Probability and statistics
• Binomial distribution$P(X=k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$Probability and statistics
• Binomial theorem$(a+b{)}^n=\sum _{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)a^{n-k}{b}^k$Algebra

C

• Chain rule$\frac{dy}{dx}=\frac{dy}{du}·\frac{du}{dx}$Calculus basics
• Change of base formula${\log }_{b}x=\frac{\ln x}{\ln b}=\frac{\log x}{\log b}$Algebra
• Circumference of a circle$C=2\pi r$Geometry
• Combinations$\left(\genfrac{}{}{0ex}{}{n}{r}\right)=\frac{n!}{r!(n-r)!}$Probability and statistics
• Compound interest$A=P{\left(1+\frac{r}{n}\right)}^{nt}$Finance maths
• Conditional probability$P(A\mid B)=\frac{P(A\cap B)}{P(B)}$Probability and statistics
• Continuous compounding$A=P{e}^{rt}$Finance maths

D

• Definite integral (area under a curve)${\int }_a^{b}f(x)dx=F(b)-F(a)$Calculus basics
• Definition of a logarithm$y={\log }_{b}x\iff b^y=x$Algebra
• Degrees and radians$\text{radians}=\text{degrees}\times \frac{\pi }{180}$Trigonometry
• Derivatives of eˣ and ln x$\frac{d}{dx}{e}^{kx}=k{e}^{kx},\frac{d}{dx}\ln x=\frac{1}{x}$Calculus basics
• Derivatives of sin and cos$\frac{d}{dx}\sin (kx)=k\cos (kx),\frac{d}{dx}\cos (kx)=-k\sin (kx)$Calculus basics
• Difference of two squares$a^2-b^2=(a+b)(a-b)$Algebra
• Difference quotient$f\prime (x)=\underset{h\to 0}{lim}\frac{f(x+h)-f(x)}{h}$Calculus basics
• Discriminant$D=b^2-4ac$Algebra
• Displacement from average velocity$s=\frac{u+v}{2}t$Motion (kinematics)
• Displacement: s = ut + ½at²$s=ut+\frac{1}{2}a{t}^2$Motion (kinematics)
• Distance formula$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$Coordinate geometry
• Distance from a point to a line$d=\frac{|a{x}_0+b{y}_0+c|}{\sqrt{a^2+b^2}}$Coordinate geometry
• Double-angle formulas$\sin 2\theta =2\sin \theta \cos \theta ,\cos 2\theta ={\cos }^2\theta -{\sin }^2\theta$Trigonometry

E

• Effective annual rate$r_{\text{eff}}={\left(1+\frac{r}{n}\right)}^n-1$Finance maths
• Equation of a circle$(x-h{)}^2+(y-k{)}^2=r^2$Coordinate geometry
• Equation of a tangent line$y=f(x_0)+f\prime (x_0)(x-x_0)$Calculus basics
• Expected value$E(X)=\sum x_i{p}_i$Probability and statistics
• Exponential growth and decay$N=N_0{e}^{kt}$Algebra

F

• Final velocity: v = u + at$v=u+at$Motion (kinematics)
• Future value of an annuity$FV=PMT·\frac{(1+i{)}^N-1}{i}$Finance maths

G

• Geometric sequence (n-th term)$a_n=a_1{r}^{n-1}$Sequences and series
• Geometric series (sum)$S_n=\frac{a_1\left(1-r^n\right)}{1-r},r\ne 1$Sequences and series

H

• Half-life formula$N=N_0{\left(\frac{1}{2}\right)}^{t/T}$Algebra
• Heron's formula$A=\sqrt{s(s-a)(s-b)(s-c)},s=\frac{a+b+c}{2}$Geometry

I

• Infinite geometric series (sum to infinity)Sequences and series
• Integral of eᵏˣ$\int e^{kx}dx=\frac{1}{k}{e}^{kx}+C$Calculus basics
• Interior angles of a polygon$S=(n-2)\times {180}^{°}$Geometry

L

• Law of cosines$c^2=a^2+b^2-2ab\cos C$Trigonometry
• Law of sines$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$Trigonometry
• Laws of exponents$a^m·a^n=a^{m+n}$Algebra
• Loan repayment formula$M=\frac{Pi}{1-(1+i{)}^{-N}},i=\frac{r}{k}$Finance maths
• Logarithm power rule${\log }_b\left(M^p\right)=p{\log }_{b}M$Algebra
• Logarithm product rule${\log }_b(MN)={\log }_{b}M+{\log }_{b}N$Algebra
• Logarithm quotient rule${\log }_b\left(\frac{M}{N}\right)={\log }_{b}M-{\log }_{b}N$Algebra

M

• Margin of error (proportion)$E=z^{\ast }\sqrt{\frac{\widehat{p}(1-\widehat{p})}{n}}$Probability and statistics
• Mean (average)$\overline{x}=\frac{\sum _{i=1}^n{x}_i}{n}$Probability and statistics
• Midpoint formula$M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$Coordinate geometry
• Multiplication rule for independent events (and)$P(A\cap B)=P(A)·P(B)$Probability and statistics

N

• Normal distribution probabilityProbability and statistics

P

• Percentage change$p=\frac{V_2-V_1}{V_1}\times 100\%$Algebra
• Permutations$P(n,r)=\frac{n!}{(n-r)!}$Probability and statistics
• Point-slope form$y-y_1=m(x-x_1)$Coordinate geometry
• Power rule for derivatives$\frac{d}{dx}\left(a{x}^n\right)=na{x}^{n-1}$Calculus basics
• Power rule for integrals$\int x^{n}dx=\frac{x^{n+1}}{n+1}+C,n\ne -1$Calculus basics
• Present value$PV=\frac{FV}{(1+r{)}^t}$Finance maths
• Present value of an annuity$PV=PMT·\frac{1-(1+i{)}^{-N}}{i}$Finance maths
• Product rule$(uv)\prime =u\prime v+uv\prime$Calculus basics
• Projectile range$R=\frac{v^2\sin 2\theta }{g}$Motion (kinematics)
• Pythagorean identity${\sin }^2\theta +{\cos }^2\theta =1$Trigonometry
• Pythagorean theorem$a^2+b^2=c^2$Geometry

Q

• Quadratic formula$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$Algebra
• Quotient rule$\left(\frac{u}{v}\right)\prime =\frac{u\prime v-uv\prime }{v^2}$Calculus basics

R

• Rule of 72$t\approx \frac{72}{100r}$Finance maths

S

• Simple interest$I=Prt$Finance maths
• Sine wave (amplitude, period and phase)$y=A\sin \left(B(x-C)\right)+D$Trigonometry
• Slope formula$m=\frac{y_2-y_1}{x_2-x_1}$Coordinate geometry
• Slope-intercept form$y=mx+b$Coordinate geometry
• Slopes of perpendicular lines$m_1·m_2=-1\iff m_2=-\frac{1}{m_1}$Coordinate geometry
• SOH CAH TOA (right-triangle trigonometry)$\sin \theta =\frac{\text{opposite}}{\text{hypotenuse}},\cos \theta =\frac{\text{adjacent}}{\text{hypotenuse}},\tan \theta =\frac{\text{opposite}}{\text{adjacent}}$Trigonometry
• Speed, distance and time$v=\frac{d}{t}$Motion (kinematics)
• Standard deviation$\sigma =\sqrt{\frac{\sum (x_i-\mu {)}^2}{N}}$Probability and statistics
• Sum of the first n squares$1^2+2^2+\cdots +n^2=\frac{n(n+1)(2n+1)}{6}$Sequences and series
• Sum of the first n whole numbers$1+2+3+\cdots +n=\frac{n(n+1)}{2}$Sequences and series
• Surface area of a box (cuboid)$S=2(lw+lh+wh)$Geometry
• Surface area of a cone$S=\pi r^2+\pi rl$Geometry
• Surface area of a cylinder$S=2\pi r^2+2\pi rh$Geometry
• Surface area of a sphere$S=4\pi r^2$Geometry

V

• Velocity and displacement: v² = u² + 2as$v^2=u^2+2as$Motion (kinematics)
• Vertex of a parabola$h=-\frac{b}{2a},k=c-\frac{b^2}{4a}$Algebra
• Volume of a box (cuboid)$V=lwh$Geometry
• Volume of a cone$V=\frac{1}{3}\pi r^{2}h$Geometry
• Volume of a cylinder$V=\pi r^{2}h$Geometry
• Volume of a pyramid$V=\frac{1}{3}Bh$Geometry
• Volume of a sphere$V=\frac{4}{3}\pi r^3$Geometry

Z

• Z-score (standard score)$z=\frac{x-\mu }{\sigma }$Probability and statistics