IB Math AI HL — Formula Booklet
Every formula printed in your official IB formula booklet, organised by topic. Knowing whichformulas you're given frees up mental space to memorise what isn't here.
★ Must memorise — NOT in the booklet
- • Rounding & significant figures rules
- • Laws of logarithms (log(ab) = log a + log b; log(aⁿ) = n log a; etc.)
- • Domain and range reasoning
- • Increasing / decreasing function tests from derivatives
- • Interpolation vs extrapolation distinction
- • Spearman's rank correlation formula — not in the booklet despite being assessed
- • Voronoi diagram construction steps
GDC workflows (Chi-squared, Finance Solver, linear regression) do not require you to recall a formula — your calculator computes them.
Prior Learning
Distance between two points (2D)
3.1
Straight-line distance between two points in the coordinate plane
Midpoint of a line segment
3.1
Coordinates of the midpoint between two points
Area of a triangle
3.2
Area = ½ × base × perpendicular height
Area of a parallelogram
Area = base × perpendicular height
Area of a trapezoid
Area = ½ × (sum of parallel sides) × height
Circumference of a circle
Perimeter of a circle
Area of a circle
Area enclosed by a circle of radius r
Volume of a cuboid
Volume = length × width × height
Volume of a cylinder
Volume of a right circular cylinder
Volume of a prism
Volume = cross-sectional area × length/height
Area of curved surface of a cylinder
Lateral surface area of a cylinder with radius r and height h
Topic 1: Number & Algebra
Arithmetic sequence — nth term
1.2
The nth term of an arithmetic sequence with first term u₁ and common difference d
Arithmetic series — sum of n terms
1.2
Sum of the first n terms of an arithmetic series
Geometric sequence — nth term
1.3
The nth term of a geometric sequence with first term u₁ and common ratio r
Geometric series — sum of n terms
1.3
Sum of the first n terms of a geometric series (r ≠ 1)
Compound interest
1.4, 1.7
Future value of an investment with annual interest rate r%, compounded k times per year for n years
Logarithm definition
1.5
Definition connecting exponential and logarithm forms
Percentage error
1.6
Percentage error where vA is approximate value and vE is exact value
Sum of an infinite geometric sequence
1.11
Converges only when |r| < 1
Complex numbers — forms
1.12, 1.13
Cartesian, polar and exponential forms
De Moivre's theorem
1.13
Powers of a complex number in polar form
Determinant & inverse of a 2×2 matrix
1.14
Inverse exists when the determinant is non-zero
Eigenvalues & eigenvectors
1.15
Characteristic equation; diagonalisation for powers of A
Topic 2: Functions
Equations of a straight line
2.1
Standard forms of a straight line
Gradient between two points
2.1
Gradient (slope) of the line through two points
Equation of a line (point-slope)
2.1
Line through point (x₁, y₁) with gradient m
Axis of symmetry of a quadratic
2.5
x-coordinate of the vertex of y = ax² + bx + c
Topic 3: Geometry & Trigonometry
Distance between two points (3D)
3.1
Straight-line distance between two points in 3D space
Midpoint in 3D
3.1
Midpoint of a line segment in 3D space
Volume of a right pyramid
3.1
Volume of a right pyramid with base area A and height h
Volume of a right cone
3.1
Volume of a right cone with radius r and height h
Area of curved surface of a cone
3.1
Lateral surface area of a cone with radius r and slant height l
Volume of a sphere
3.1
Volume of a sphere with radius r
Surface area of a sphere
3.1
Total surface area of a sphere with radius r
Sine rule
3.2
Relates each side of a triangle to the sine of the opposite angle
Cosine rule — finding a side
3.2
Finds side c given sides a, b and included angle C
Cosine rule — finding an angle
3.2
Finds angle C given all three sides a, b, c
Area of a triangle (trig)
3.2
Area of a triangle given two sides and the included angle
Arc length
3.4
Length of an arc with central angle θ° in a circle of radius r
Area of a sector
3.4
Area of a sector with central angle θ° in a circle of radius r
Radian measure
3.7
Arc length and sector area with θ in radians
Scalar (dot) product
3.13
θ is the angle between the vectors
Vector (cross) product
3.13
Magnitude = area of the parallelogram
Vector equation of a line
3.11
Point a, direction b
Topic 4: Statistics & Probability
Interquartile range
4.2
Spread of the middle 50% of data
Mean of a frequency distribution
4.3
Weighted mean where fᵢ is the frequency of value xᵢ
Probability of an event
4.5
Probability = (favourable outcomes) / (total outcomes) in a uniform sample space
Complementary events
4.5
The probabilities of an event and its complement sum to 1
Addition rule
4.6
Probability of A or B (or both)
Mutually exclusive events
4.6
When A and B cannot both occur
Conditional probability
4.6
Probability of A given that B has already occurred
Independent events
4.6
If A and B are independent, their joint probability is the product of their probabilities
Expected value of a discrete distribution
4.7
Long-run average value of a discrete random variable
Binomial distribution — mean
4.8
Expected value of X ~ B(n, p)
Binomial distribution — variance
4.8
Variance of X ~ B(n, p)
Linear transformation of a random variable
4.14
Effect of a linear change on mean and variance
Poisson distribution
4.17
Mean equals variance equals m
Confidence interval for a mean
4.16
z from the required confidence level
Markov chains
4.19
Transition matrix T; steady (long-run) state vector s
Topic 5: Calculus
Power rule (differentiation)
5.1, 5.3
Derivative of a power function
Integration of a power function
5.5
Indefinite integral of xⁿ (n ≠ −1)
Area under a curve (definite integral)
5.5
Signed area between f(x) and the x-axis from a to b
Trapezoidal rule
5.8
Numerical approximation of a definite integral using trapezoids
Standard derivatives (HL)
5.9
Plus the product, quotient and chain rules
Standard integrals (HL)
5.11
With integration by substitution / reverse chain
Volume of revolution
5.12
Rotating about the x-axis or the y-axis
Euler's method
5.16, 5.18
Step-by-step numerical solution of a differential equation
Source: IB Diploma Programme Mathematics: applications and interpretation — HL formula booklet (first examinations 2021). Always verify against your official IB materials.