Area of a parallelogram
[A = b h]
Base times perpendicular height.
Area of a triangle
[A = \tfrac{1}{2} b h]
Half of base times perpendicular height.
Area of a trapezoid
[A = \tfrac{1}{2}(a + b)h]
Average of the two parallel sides times the height between them.
Area of a circle
[A = \pi r^2]
π times the radius squared.
Circumference of a circle
[C = 2\pi r]
The distance around a circle.
Volume of a cuboid
[V = l w h]
Length × width × height.
Volume of a cylinder
[V = \pi r^2 h]
Circle area times height.
Volume of a prism
[V = A h]
Cross-section area times length.
Curved surface of a cylinder
[A = 2\pi r h]
Unroll the side into a rectangle: circumference × height.
Distance between two points
[d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}]
Pythagoras on the horizontal and vertical gaps.
Midpoint of a segment
[\left(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2}\right)]
Average the x-coordinates and the y-coordinates.
Arithmetic sequence — nth term
[u_n = u_1 + (n-1)d]
Start at the first term and add the common difference once per step.
Arithmetic series — sum
[S_n = \tfrac{n}{2}(2u_1 + (n-1)d) = \tfrac{n}{2}(u_1 + u_n)]
Average the first and last term, times the number of terms.
Geometric sequence — nth term
[u_n = u_1 r^{\,n-1}]
Multiply the first term by the ratio once per step.
Geometric series — sum
[S_n = \dfrac{u_1(r^n - 1)}{r - 1}, \quad r \ne 1]
Shortcut for adding terms that keep multiplying by r.
Sum to infinity
[S_\infty = \dfrac{u_1}{1 - r}, \quad |r| < 1]
If terms shrink, infinitely many add to a finite total.
Compound interest
[FV = PV\left(1 + \dfrac{r}{100k}\right)^{kn}]
PV = start; r% = yearly rate; k = compounds per year; n = years.
Exponent ↔ logarithm
[a^x = b \iff x = \log_a b]
Logs undo exponentials; use a log when the unknown is a power.
Laws of logarithms
[\begin{aligned}\log_a(xy)&=\log_a x+\log_a y\\\log_a\tfrac{x}{y}&=\log_a x-\log_a y\\\log_a x^m&=m\log_a x\end{aligned}]
Products → sums, quotients → differences, powers come out front.
Change of base
[\log_a x = \dfrac{\log_b x}{\log_b a}]
Rewrite a log in a base your calculator has (10 or e).
Binomial theorem
[(a+b)^n = a^n + \binom{n}{1}a^{n-1}b + \dots + b^n]
Expands a bracket to a power; each term chooses r b's via C(n,r).
Axis of symmetry of a parabola
[x = -\dfrac{b}{2a}]
The vertical line through the vertex — halfway between the roots.
Quadratic formula
[x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}]
Solves any quadratic; the ± gives the two roots.
Discriminant
[\Delta = b^2 - 4ac]
Δ>0 → two roots; Δ=0 → one; Δ<0 → none.
Volume of a sphere
[V = \tfrac{4}{3}\pi r^3]
Grows with the cube of the radius.
Surface area of a sphere
[A = 4\pi r^2]
Exactly four circle-areas.
Volume of a cone
[V = \tfrac{1}{3}\pi r^2 h]
A third of the cylinder with the same base and height.
Curved surface of a cone
[A = \pi r l]
Uses the slant height l, not the vertical height.
Volume of a pyramid
[V = \tfrac{1}{3} A h]
A third of base area times height.
Right-angled trig (SOH-CAH-TOA)
[\begin{aligned}\sin\theta&=\tfrac{\text{opp}}{\text{hyp}}\\\cos\theta&=\tfrac{\text{adj}}{\text{hyp}}\\\tan\theta&=\tfrac{\text{opp}}{\text{adj}}\end{aligned}]
The three ratios linking an angle to the sides of a right triangle.
Sine rule
[\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}]
Pairs each side with its opposite angle.
Cosine rule
[c^2 = a^2 + b^2 - 2ab\cos C]
Pythagoras plus an angle-correction term.
Area of a triangle (two sides & angle)
[A = \tfrac{1}{2}ab\sin C]
Half the product of two sides times the sine of the angle between.
Arc length (radians)
[l = r\theta]
Radius times the angle (θ in radians).
Sector area (radians)
[A = \tfrac{1}{2}r^2\theta]
The 'pizza slice' area (θ in radians).
Pythagorean identity
[\sin^2\theta + \cos^2\theta = 1]
Lets you swap between sin and cos.
Tangent identity
[\tan\theta = \dfrac{\sin\theta}{\cos\theta}]
Definition of tan in terms of sin and cos.
Double angle identities
[\begin{aligned}\sin 2\theta&=2\sin\theta\cos\theta\\\cos 2\theta&=\cos^2\theta-\sin^2\theta\end{aligned}]
Rewrite trig of 2θ in terms of θ.
Mean from a frequency table
[\bar x = \dfrac{\sum f x}{\sum f}]
Each value weighted by how often it occurs.
Probability of an event
[P(A) = \dfrac{n(A)}{n(U)}]
Favourable outcomes over total outcomes.
Combined events (addition rule)
[P(A\cup B) = P(A) + P(B) - P(A\cap B)]
Add the chances, subtract the overlap.
Conditional probability
[P(A\mid B) = \dfrac{P(A\cap B)}{P(B)}]
Chance of A once B is known — sample space shrinks to B.
Independent events
[P(A\cap B) = P(A)\,P(B)]
Multiply when one event doesn't affect the other.
Expected value
[E(X) = \sum x\,P(X=x)]
Long-run average: outcomes weighted by probability.
Binomial distribution
[\begin{aligned}X&\sim B(n,p)\\E(X)&=np\\\operatorname{Var}(X)&=np(1-p)\end{aligned}]
n independent trials, success chance p; on average np successes.
Standardising (z-score)
[z = \dfrac{x - \mu}{\sigma}]
How many standard deviations a value is from the mean.
Pearson's correlation & regression
[\begin{aligned}y&=ax+b\;\text{(least squares)}\\-1&\le r\le 1\end{aligned}]
Line of best fit; r measures strength/direction of linear association.
Derivative — power rule
[\dfrac{d}{dx}(x^n) = n x^{\,n-1}]
Bring the power down, reduce it by one.
Standard derivatives
[\begin{aligned}\tfrac{d}{dx}\sin x&=\cos x\\\tfrac{d}{dx}e^x&=e^x\\\tfrac{d}{dx}\ln x&=\tfrac{1}{x}\end{aligned}]
The common functions and their derivatives.
Chain rule
[\dfrac{dy}{dx} = \dfrac{dy}{du}\cdot\dfrac{du}{dx}]
Differentiate the outside, times the derivative of the inside.
Product & quotient rules
[\begin{aligned}(uv)'&=u'v+uv'\\[4pt]\left(\tfrac{u}{v}\right)'&=\dfrac{u'v-uv'}{v^2}\end{aligned}]
For products and fractions of two functions.
Integration — power rule
[\int x^n\,dx = \dfrac{x^{\,n+1}}{n+1} + C, \quad n\ne -1]
Reverse of differentiating: raise the power, divide by it.
Standard integrals
[\begin{aligned}\int e^x\,dx&=e^x+C\\\int\tfrac{1}{x}\,dx&=\ln|x|+C\\\int\cos x\,dx&=\sin x+C\end{aligned}]
The common antiderivatives.
Area under a curve
[A = \int_a^b y\,dx]
A definite integral sums thin strips into a signed area.
Kinematics
[\begin{aligned}v&=\dfrac{ds}{dt}\\a&=\dfrac{dv}{dt}\\s&=\int v\,dt\end{aligned}]
Differentiate to go displacement→velocity→acceleration; integrate to reverse.
