Area of a parallelogram
[A = b h]
Base times perpendicular height.
When: Any parallelogram.
AA SLAA HLAI SLAI HL
Prior learning
Area of a triangle
[A = \tfrac{1}{2} b h]
Half of base times perpendicular height.
When: When you know a base and its height.
AA SLAA HLAI SLAI HL
Area of a trapezoid
[A = \tfrac{1}{2}(a + b)h]
Average of the two parallel sides times the height between them.
When: Four-sided shapes with one pair of parallel sides.
AA SLAA HLAI SLAI HL
Area of a circle
[A = \pi r^2]
π times the radius squared.
When: Circles and disc-shaped regions.
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Circumference of a circle
[C = 2\pi r]
The distance around a circle.
When: Perimeter of a circle.
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Volume of a cuboid
[V = l w h]
Length × width × height.
When: Rectangular boxes.
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Volume of a cylinder
[V = \pi r^2 h]
Circle area times height.
When: Cylinders / tubes.
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Volume of a prism
[V = A h]
Cross-section area times length.
When: Any solid with a constant cross-section.
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Curved surface of a cylinder
[A = 2\pi r h]
Unroll the side into a rectangle: circumference × height.
When: Surface area of the curved part.
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Distance between two points
[d = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}]
Pythagoras on the horizontal and vertical gaps.
When: Length of a segment from coordinates.
AA SLAA HLAI SLAI HL
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.
When: The point halfway between two points.
AA SLAA HLAI SLAI HL
Number & Algebra
Arithmetic sequence — nth term
[u_n = u_1 + (n-1)d]
Start at the first term and add the common difference once per step.
When: Sequences that go up/down by a fixed amount.
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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.
When: Totalling an arithmetic sequence.
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Geometric sequence — nth term
[u_n = u_1 r^{\,n-1}]
Multiply the first term by the ratio once per step.
When: Growth/decay by a fixed factor.
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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.
When: Repeated percentage growth (e.g. savings).
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Sum to infinity
[S_\infty = \dfrac{u_1}{1 - r}, \quad |r| < 1]
If terms shrink, infinitely many add to a finite total.
When: Only when |r| < 1.
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Compound interest
[FV = PV\left(1 + \dfrac{r}{100k}\right)^{kn}]
PV = start; r% = yearly rate; k = compounds per year; n = years.
When: Investments/loans compounding more than once a year.
AA SLAA HLAI SLAI HL
Percentage error
[\varepsilon = \left|\dfrac{v_A - v_E}{v_E}\right| \times 100\%]
How far an approximate value is from the exact one, as a %.
When: Rounded/measured vs true value.
AI SLAI HL
Exponent ↔ logarithm
[a^x = b \iff x = \log_a b]
Logs undo exponentials; use a log when the unknown is a power.
When: Solving for an exponent.
AA SLAA HLAI HL
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.
When: Combining/splitting logs to solve equations.
AA SLAA HLAI HL
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).
When: Evaluating e.g. log₅30.
AA SLAA HLAI HL
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).
When: Expanding powers or finding one term.
AA SLAA HL
Binomial — general term
[\binom{n}{r} a^{\,n-r} b^{\,r}]
The (r+1)th term — set the power of x to find a specific term.
When: Finding a coefficient without full expansion.
AA HL
Combinations and permutations
[\begin{aligned}\binom{n}{r}&=\dfrac{n!}{r!(n-r)!}\\[6pt]{}^nP_r&=\dfrac{n!}{(n-r)!}\end{aligned}]
C counts selections (order doesn't matter); P counts arrangements (order matters).
When: Counting problems, binomial coefficients.
AA HL
Complex number — modulus & argument
[\begin{aligned}z&=a+bi\\|z|&=\sqrt{a^2+b^2}\\\arg z&=\arctan\tfrac{b}{a}\end{aligned}]
Modulus = distance from origin; argument = angle from positive real axis.
When: Polar form, Argand diagrams.
AA HLAI HL
Complex — polar / Euler form
[z = r(\cos\theta + i\sin\theta) = r\,e^{i\theta}]
Two compact ways to write a complex number using its modulus and angle.
When: Multiplying, dividing and powering complex numbers.
AA HL
De Moivre's theorem
[(r\,\text{cis}\,\theta)^n = r^n\,\text{cis}(n\theta)]
Power the modulus, multiply the angle by n.
When: Powers and nth roots of complex numbers.
AA HL
Determinant & inverse of a 2×2 matrix
[\begin{aligned}\det\begin{pmatrix}a&b\\c&d\end{pmatrix}&=ad-bc\\[8pt]M^{-1}&=\tfrac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\end{aligned}]
Cross-multiply diagonals and subtract; inverse swaps the diagonal and negates the off-diagonal.
When: Solving systems, transformations (AI HL).
AI HL
Functions
Axis of symmetry of a parabola
[x = -\dfrac{b}{2a}]
The vertical line through the vertex — halfway between the roots.
When: Max/min of a quadratic.
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Quadratic formula
[x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}]
Solves any quadratic; the ± gives the two roots.
When: When it won't factorise nicely.
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Discriminant
[\Delta = b^2 - 4ac]
Δ>0 → two roots; Δ=0 → one; Δ<0 → none.
When: How many real roots a quadratic has.
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Geometry & Trigonometry
Volume of a sphere
[V = \tfrac{4}{3}\pi r^3]
Grows with the cube of the radius.
When: Balls, domes.
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Surface area of a sphere
[A = 4\pi r^2]
Exactly four circle-areas.
When: Outer area of a sphere.
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Volume of a cone
[V = \tfrac{1}{3}\pi r^2 h]
A third of the cylinder with the same base and height.
When: Cones.
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Curved surface of a cone
[A = \pi r l]
Uses the slant height l, not the vertical height.
When: Surface of the sloping part.
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Volume of a pyramid
[V = \tfrac{1}{3} A h]
A third of base area times height.
When: Pyramids.
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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.
When: Right-angled triangles.
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Sine rule
[\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}]
Pairs each side with its opposite angle.
When: A side + its opposite angle are known.
AA SLAA HLAI SLAI HL
Cosine rule
[c^2 = a^2 + b^2 - 2ab\cos C]
Pythagoras plus an angle-correction term.
When: Two sides + included angle, or all three sides.
AA SLAA HLAI SLAI HL
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.
When: No perpendicular height available.
AA SLAA HLAI SLAI HL
Arc length (degrees)
[l = \dfrac{\theta}{360}\times 2\pi r]
The fraction θ/360 of the full circumference.
When: AI — angles in degrees.
AI SLAI HL
Sector area (degrees)
[A = \dfrac{\theta}{360}\times \pi r^2]
The fraction θ/360 of the full circle area.
When: AI — angles in degrees.
AI SLAI HL
Arc length (radians)
[l = r\theta]
Radius times the angle (θ in radians).
When: AA — angles in radians.
AA SLAA HL
Sector area (radians)
[A = \tfrac{1}{2}r^2\theta]
The 'pizza slice' area (θ in radians).
When: AA — angles in radians.
AA SLAA HL
Pythagorean identity
[\sin^2\theta + \cos^2\theta = 1]
Lets you swap between sin and cos.
When: Given one of sinθ/cosθ, find the other.
AA SLAA HL
Tangent identity
[\tan\theta = \dfrac{\sin\theta}{\cos\theta}]
Definition of tan in terms of sin and cos.
When: Simplifying / solving trig equations.
AA SLAA HL
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.
When: Simplifying, integrating, exact values.
AA SLAA HL
Compound angle identities
[\begin{aligned}\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\end{aligned}]
Break the sin/cos of a sum into parts.
When: Proofs, exact values (AA HL).
AA HL
Vector magnitude & scalar product
[\begin{aligned}|\mathbf{v}|&=\sqrt{v_1^2+v_2^2+v_3^2}\\\mathbf{a}\cdot\mathbf{b}&=|\mathbf{a}||\mathbf{b}|\cos\theta\end{aligned}]
Magnitude = length; the dot product gives the angle between vectors.
When: Angles, projections, perpendicularity.
AA HLAI HL
Vector (cross) product
[|\mathbf{a}\times\mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta]
Gives a vector perpendicular to both; its size is the parallelogram area.
When: Normals, areas, planes (AA HL).
AA HL
Statistics & Probability
Mean from a frequency table
[\bar x = \dfrac{\sum f x}{\sum f}]
Each value weighted by how often it occurs.
When: Grouped or repeated data.
AA SLAA HLAI SLAI HL
Probability of an event
[P(A) = \dfrac{n(A)}{n(U)}]
Favourable outcomes over total outcomes.
When: Equally-likely outcomes.
AA SLAA HLAI SLAI HL
Combined events (addition rule)
[P(A\cup B) = P(A) + P(B) - P(A\cap B)]
Add the chances, subtract the overlap.
When: P(A or B).
AA SLAA HLAI SLAI HL
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.
When: 'Given that…' problems.
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Independent events
[P(A\cap B) = P(A)\,P(B)]
Multiply when one event doesn't affect the other.
When: Both happen; also the test for independence.
AA SLAA HLAI SLAI HL
Expected value
[E(X) = \sum x\,P(X=x)]
Long-run average: outcomes weighted by probability.
When: Fair games, average payoff.
AA SLAA HLAI SLAI HL
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.
When: Fixed number of yes/no trials.
AA SLAA HLAI SLAI HL
Standardising (z-score)
[z = \dfrac{x - \mu}{\sigma}]
How many standard deviations a value is from the mean.
When: Normal-distribution problems by hand.
AA SLAA HL
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.
When: Bivariate data, prediction.
AA SLAA HLAI SLAI HL
Poisson distribution
[P(X=r) = \dfrac{e^{-\lambda}\lambda^{\,r}}{r!}]
Counts of random events at average rate λ.
When: Events per interval (calls/hour etc.).
AA HLAI HL
Chi-squared test statistic
[\chi^2 = \sum \dfrac{(f_o - f_e)^2}{f_e}]
Compares observed with expected frequencies.
When: Tests of independence / goodness-of-fit (AI).
AI SLAI HL
Bayes' theorem
[P(A\mid B) = \dfrac{P(A)P(B\mid A)}{P(B)}]
Reverses a conditional probability using the overall rate.
When: Test-accuracy / diagnostic problems (HL).
AA HLAI HL
Calculus
Derivative — power rule
[\dfrac{d}{dx}(x^n) = n x^{\,n-1}]
Bring the power down, reduce it by one.
When: Differentiating powers of x.
AA SLAA HLAI SLAI HL
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.
When: Differentiating trig/exponential/log (AA).
AA SLAA HL
Chain rule
[\dfrac{dy}{dx} = \dfrac{dy}{du}\cdot\dfrac{du}{dx}]
Differentiate the outside, times the derivative of the inside.
When: Function inside a function.
AA SLAA HL
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.
When: Differentiating e.g. x²eˣ or x/(x+1).
AA SLAA HL
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.
When: Integrating powers of x (don't forget +C).
AA SLAA HLAI HL
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.
When: Integrating exponential/log/trig (AA).
AA SLAA HL
Area under a curve
[A = \int_a^b y\,dx]
A definite integral sums thin strips into a signed area.
When: Area, and (with velocity) distance.
AA SLAA HLAI HL
Volume of revolution
[V = \pi\int_a^b y^2\,dx]
Rotate a region about the x-axis and sum disc volumes.
When: Solids of revolution (AA HL).
AA HL
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.
When: Motion problems.
AA SLAA HLAI HL
Trapezoidal rule
[\int_a^b y\,dx \approx \tfrac{h}{2}\big(y_0 + y_n + 2(y_1+\dots+y_{n-1})\big)]
Estimate area with trapezia of width h.
When: Approximating an integral (AI).
AI SLAI HL
Integration by parts
[\int u\,dv = uv - \int v\,du]
Swaps a hard integral for an easier one; choose u to simplify when differentiated.
When: Products like x·eˣ (AA HL).
AA HL
Maclaurin series
[f(x) = f(0) + f'(0)x + \dfrac{f''(0)}{2!}x^2 + \dots]
Approximates a function near 0 as an infinite polynomial.
When: Series approximations (AA HL).
AA HL