IB Math AA 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
- • Exact values of sin, cos and tan for 0°, 30°, 45°, 60°, 90°
- • Laws of indices (aᵐ·aⁿ = aᵐ⁺ⁿ, etc.)
- • Completing the square and the vertex form of a quadratic
- • Graph transformations (translations, stretches, reflections)
- • The relationship sin²θ + cos²θ = 1 rearranged for proofs
- • Differentiation from first principles (the limit definition)
On Paper 2 the GDC computes numeric work (solving equations, integrals, distributions) — you do not need to recall those methods as formulas.
Prior Learning
Area of a parallelogram
Area = base × perpendicular height
Area of a triangle
Area = ½ × base × perpendicular height
Area of a trapezoid
Area = ½ × (sum of parallel sides) × height
Area of a circle
Area enclosed by a circle of radius r
Circumference of a circle
Perimeter of 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
Curved surface area of a cylinder
Lateral surface area of a cylinder
Distance between two points (2D)
3.1
Straight-line distance between two points in the plane
Midpoint of a line segment (2D)
2.1, 3.1
Coordinates of the midpoint between two points
Topic 1: Number & Algebra
Arithmetic sequence — nth term
1.2
nth term 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 sequence
Geometric sequence — nth term
1.3
nth term with first term u₁ and common ratio r
Geometric series — sum of n terms
1.3
Sum of the first n terms of a finite geometric sequence
Sum of an infinite geometric sequence
1.8
Sum to infinity of a convergent geometric series
Compound interest
1.4
FV future value, PV present value, n years, k compounding periods/year, r% nominal annual rate
Exponents and logarithms
1.5, 1.7
Definition linking an exponent to a logarithm
Laws of logarithms
1.7
Product, quotient, power and change-of-base laws
Exponential and logarithmic functions
1.7
Converting between bases using e and natural logarithms
Binomial theorem
1.9
Expansion of (a + b)ⁿ, where n ∈ ℕ
Combinations (binomial coefficient)
1.9
Number of ways to choose r items from n (ⁿCᵣ)
Permutations
1.10
Number of ordered arrangements of r items chosen from n
Combinations
1.10
Number of unordered selections of r items chosen from n
Extension of the binomial theorem (n ∈ ℚ)
1.10
Extended binomial expansion for any rational power n (valid for |b/a| < 1)
Complex numbers — Cartesian, polar & exponential form
1.12, 1.13
The three equivalent forms of a complex number (r = |z| is the modulus, θ = arg z is the argument)
De Moivre's theorem
1.14
Powers (and roots) of a complex number in polar/exponential form
Topic 2: Functions
Equations of a straight line
2.1
Slope-intercept, general and point-slope forms
Gradient formula
2.1
Gradient between two points
Axis of symmetry of a quadratic
2.6
Equation of the axis of symmetry of a parabola
Solutions of a quadratic equation
2.7
Roots of ax² + bx + c = 0
Discriminant
2.7
Determines the number of real roots of a quadratic
Sum and product of the roots of polynomial equations
2.12
For aₙxⁿ + … + a₁x + a₀ = 0: sum of roots and product of roots
Topic 3: Geometry & Trigonometry
Distance between two points (3D)
3.1
Straight-line distance between two points in space
Midpoint of a line segment (3D)
3.1
Coordinates of the midpoint between two points in space
Volume of a right pyramid
3.1
A is the area of the base, h is the height
Volume of a right cone
3.1
r is the radius, h is the height
Curved surface area of a cone
3.1
r is the radius, l is the slant height
Volume of a sphere
3.1
r is the radius
Surface area of a sphere
3.1
r is the radius
Sine rule
3.2
Relates sides and opposite angles in any triangle
Cosine rule
3.2
Finds a third side or an angle in any triangle
Area of a triangle
3.2
Area from two sides and the included angle
Length of an arc
3.4
r is the radius, θ in radians
Area of a sector
3.4
r is the radius, θ in radians
Identity for tan θ
3.5
Definition of tangent in terms of sine and cosine
Pythagorean identity
3.6
Fundamental trigonometric identity
Double angle identities
3.6
Identities for sin 2θ and cos 2θ
Reciprocal trig identities
3.9
The two further Pythagorean identities
Compound angle identities
3.10
Sums and differences of angles
Scalar (dot) product
3.13
Two forms of the scalar product (θ is the angle between the vectors)
Vector equation of a line
3.14
Point a, direction b, parameter λ
Vector (cross) product
3.16
Magnitude of the vector product = area of the parallelogram
Vector & Cartesian equation of a plane
3.17
n is the normal vector; d = a·n
Topic 4: Statistics & Probability
Interquartile range
4.2
Spread of the middle 50% of the data
Mean of a set of data
4.3
Mean from a frequency distribution
Probability of an event A
4.5
Favourable outcomes ÷ total outcomes
Complementary events
4.5
The probabilities of an event and its complement sum to 1
Combined events
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 occurred
Independent events
4.6
When the occurrence of one does not affect the other
Expected value of a discrete random variable
4.7
Long-run average value of X
Binomial distribution X ~ B(n, p)
4.8
Mean and variance of a binomial distribution
Standardized normal variable
4.12
z-score: how many standard deviations x is from the mean
Bayes' theorem
4.13
Reversing conditional probabilities
Continuous random variables
4.14
Expectation and variance from a probability density function f(x)
Topic 5: Calculus
Derivative of xⁿ
5.3
Power rule for differentiation
Integral of xⁿ
5.5
Power rule for integration
Area between a curve and the x-axis
5.5, 5.11
Definite integral as the area under y = f(x)
Derivatives of sin x, cos x, eˣ and ln x
5.6
Standard derivatives given in the booklet
Chain rule
5.6
Differentiating a composite function
Product rule
5.6
Differentiating a product of two functions
Quotient rule
5.6
Differentiating a quotient of two functions
Acceleration
5.9
Acceleration as the rate of change of velocity
Distance and displacement from velocity
5.9
Integrate speed for distance, velocity for displacement
Standard integrals
5.10
Standard integrals given in the booklet
Derivative from first principles
5.12
The limit definition of the derivative
Standard derivatives (HL)
5.15
Derivatives of tan, aˣ, arcsin, arccos, arctan, sec, csc, cot, logₐx
Integration by parts
5.16
Reverse of the product rule
Volume of revolution
5.17
Rotating a region about the x-axis or the y-axis
Maclaurin series
5.19
Power series about x = 0
Source: IB Diploma Programme Mathematics: analysis and approaches — HL formula booklet (first examinations 2021). Always verify against your official IB materials.