IB Maths AI HL Formula Booklet — Applications & Interpretation HL

Quick answer: yes, you get a clean copy of the IB Maths AI HL formula booklet in all three papers. SL and HL have separate booklets, and the HL booklet includes the SL formulas; the 2026 exams use the current editions (first assessment 2021). Your school can give you a copy to revise with: the IB publishes it only to schools on its Programme Resource Centre, and there is no public copy on ibo.org (we don't host it either; the course is on the IB DP Mathematics page). It does not have everything, so also learn the 46 results it leaves out.

The 18 formulas that HL adds for IB Mathematics Applications & Interpretation HL, grouped by syllabus topic and written in the same notation as the IB formula booklet. You also need every formula in the IB Maths AI SL formula booklet: they are further down this page, and the free PDF has both on one page.

Download the free PDF (SL + HL) Practise these formulas: free questions

Updated November 2026

What's in the IB Maths AI HL formula booklet?

There is a separate SL booklet and HL booklet for AI. This table is for the AI HL booklet, which includes the SL formulas as well as the HL ones. Topic by topic, this is what it prints and the main results it leaves out (★), which you must learn.

IB Maths AI HL formula booklet by topic: printed, and ★ not printed
Topic	Printed in the booklet	★ Not printed: learn these
Prior learning	Areas · Circumference; cylinder · Cuboid; prism · Distance; midpoint	Pythagoras; SOHCAHTOA; indices
Number & algebra	Arithmetic · Geometric · Sum to infinity, $\| r \| < 1$ · Compound interest · Percentage error · Exponents & logs · and 5 more	Standard form · Modulus, argument · Products · Sinusoids · 3×3 determinant · Matrix rules · and 2 more
Functions & models	Lines · Quadratic	Perpendicular · Models · Logistic model · Linearising · Composite; inverse · Transformations
Geometry & trigonometry	3D distance ( $Δ x = x_{1} - x_{2}$ , …) · 3D midpoint · Pyramid; cone; sphere · Sine, cosine rules; area · Arc; sector, degrees · Arc; sector, radians · and 9 more	Radians; bearings · Perpendicular bisectors & Voronoi · Area of image · Composite transformation · Motion with constant velocity · Unit vector; perpendicular · and 7 more
Statistics & probability	IQR; mean · Probability · Combined events · Independent · Conditional · Expected value · and 6 more	Outliers · Variance · Binomial pmf · Poisson $X \sim Po (m)$ · Sum of independent Poissons · Sample mean (CLT, $n \geq 30$ or $X$ normal) · and 5 more
Calculus	Derivatives · Chain, product, quotient · Integrals · More integrals · Areas · Volumes of revolution · and 5 more	Tangent; normal · Max / min; inflexion · Related rates · Linear inside; by inspection · Separable · Phase portraits · and 1 more

Every row is on our free one-page IB Maths AI HL formula sheet (PDF), with ★ on the ones to learn.

Everything in SL AI, plus…

Complex numbers — Cartesian ★ not in the booklet: $z = a + b i, | z | = \sqrt{a^{2} + b^{2}}, \arg z = \arctan (\frac{b}{a})$
Modulus of a complex number ★ not in the booklet: $| z_{1} z_{2} | = | z_{1} | \cdot | z_{2} |, | \frac{z_{1}}{z_{2}} | = \frac{| z_{1} |}{| z_{2} |}$

Topic 3 (HL only) — Geometry & Trig extension

Vector dot product: $\vec{a} \cdot \vec{b} = | \vec{a} | | \vec{b} | \cos θ = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}$
Vector magnitude: $| \vec{a} | = \sqrt{a_{1}^{2} + a_{2}^{2} + a_{3}^{2}}$
Angle between two vectors: $\cos θ = \frac{\vec{a} \cdot \vec{b}}{| \vec{a} | | \vec{b} |}$
Vector equation of a line: $\vec{r} = \vec{a} + t \vec{d}$
Matrix multiplication (2×2): $(\begin{matrix} a & b \\ c & d \end{matrix}) (\begin{matrix} e & f \\ g & h \end{matrix}) = (\begin{matrix} a e + b g & a f + b h \\ c e + d g & c f + d h \end{matrix})$
Determinant (2×2): $det (\begin{matrix} a & b \\ c & d \end{matrix}) = a d - b c$
Inverse of a 2×2 matrix: $M^{- 1} = \frac{1}{det M} (\begin{matrix} d & - b \\ - c & a \end{matrix})$
Eigenvalue equation ★ not in the booklet: $M \vec{v} = λ \vec{v} ⟹ det (M - λ I) = 0$

Practise Topic 3: free HL AI questions with mark schemes →

Using the booklet: a worked example

$\frac{d y}{d x} = x + y$ with $y = 1$ when $x = 0$ . Use Euler's method with step $h = 0.1$ to estimate $y$ when $x = 0.2$ .

From the booklet, $y_{n + 1} = y_{n} + h \times f (x_{n}, y_{n})$ and $x_{n + 1} = x_{n} + h$ , here with $f (x, y) = x + y$ .
Step 1: $y_{1} = 1 + 0.1 (0 + 1) = 1.1$ at $x_{1} = 0.1$ .
Step 2: $y_{2} = 1.1 + 0.1 (0.1 + 1.1) = 1.22$ at $x_{2} = 0.2$ .
The exact solution is $y = 2 e^{x} - x - 1$ , which gives $1.2428 \dots$ : Euler's method underestimates here because the curve is concave up.

Euler's method explained · Free HL AI differential equations and Euler's method questions with mark schemes

Topic 4 (HL only) — Statistics extension

Poisson distribution: $P (X = r) = \frac{e^{- λ} λ^{r}}{r!}$
Mean & variance of Poisson: $E (X) = λ, Var (X) = λ$
Confidence interval for a mean (large n) ★ not in the booklet: $\bar{x} \pm z^{*} \cdot \frac{σ}{\sqrt{n}}$
Chi-squared test statistic: $χ_{calc}^{2} = \sum \frac{(f_{o} - f_{e})^{2}}{f_{e}}$
Transition matrix (Markov chain) ★ not in the booklet: ${\vec{s}}_{n + 1} = T {\vec{s}}_{n}$

Practise Topic 4: free HL AI questions with mark schemes →

Topic 5 (HL only) — Calculus extension

Euler's method: $y_{n + 1} = y_{n} + h f (x_{n}, y_{n}), x_{n + 1} = x_{n} + h$
Separable differential equation ★ not in the booklet: $\frac{d y}{d x} = f (x) g (y) ⟹ \int \frac{1}{g (y)} d y = \int f (x) d x$
Trapezoidal rule (refresher): $A \approx \frac{h}{2} (y_{0} + y_{n} + 2 \sum_{i = 1}^{n - 1} y_{i})$

Practise Topic 5: free HL AI questions with mark schemes →

The SL formulas (also in the HL booklet)

HL papers use every IB Maths AI SL formula as well as the HL ones above. IB Maths AI SL formula booklet page.

Prior learning

Area of a parallelogram: $A = b \cdot h$
Area of a triangle: $A = \frac{1}{2} (b \cdot h)$
Area of a trapezium: $A = \frac{1}{2} (a + b) h$
Area of a circle: $A = π r^{2}$
Circumference of a circle: $C = 2 π r$
Volume of a cuboid: $V = l \cdot w \cdot h$
Volume of a cylinder: $V = π r^{2} h$
Volume of a prism: $V = A \cdot h (A = cross-section area)$
Area of curved surface of cylinder: $A = 2 π r h$
Distance between two points: $d = \sqrt{(x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2}}$
Midpoint of two points: $(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$

Topic 1 — Number & Algebra

The n-th term of an arithmetic sequence: $u_{n} = u_{1} + (n - 1) d$
Sum of n terms of an arithmetic sequence: $S_{n} = \frac{n}{2} (2 u_{1} + (n - 1) d) = \frac{n}{2} (u_{1} + u_{n})$
The n-th term of a geometric sequence: $u_{n} = u_{1} \cdot r^{n - 1}$
Sum of n terms of a geometric sequence: $S_{n} = \frac{u_{1} (r^{n} - 1)}{r - 1} = \frac{u_{1} (1 - r^{n})}{1 - r}, r \neq 1$
Compound interest: $F V = P V {(1 + \frac{r}{100 k})}^{k n}$
Exponents: $a^{p} \cdot a^{q} = a^{p + q}, \frac{a^{p}}{a^{q}} = a^{p - q}, (a^{p})^{q} = a^{p q}$
Logarithms — power law: $\log_{a} (x^{n}) = n \log_{a} x$
% error: $ε = | \frac{v_{A} - v_{E}}{v_{E}} | \times 100 %$

Topic 2 — Functions

Equation of a straight line: $y = m x + c or a x + b y + d = 0$
Gradient of a straight line: $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$
Axis of symmetry of a quadratic: $x = - \frac{b}{2 a}$

Topic 3 — Geometry & Trigonometry

Length of an arc: $ℓ = 2 π r \cdot \frac{θ}{360^{\circ}}$
Area of a sector: $A = π r^{2} \cdot \frac{θ}{360^{\circ}}$
Surface area of a sphere: $A = 4 π r^{2}$
Volume of a sphere: $V = \frac{4}{3} π r^{3}$
Volume of a cone: $V = \frac{1}{3} π r^{2} h$
Curved surface area of a cone: $A = π r l$
Volume of a pyramid: $V = \frac{1}{3} A_{base} \cdot h$
Right-angled triangle (SOHCAHTOA): $\sin θ = \frac{O}{H}, \cos θ = \frac{A}{H}, \tan θ = \frac{O}{A}$
Sine rule: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
Cosine rule: $c^{2} = a^{2} + b^{2} - 2 a b \cos C$
Area of a triangle (SAS): $A = \frac{1}{2} a b \sin C$

Topic 4 — Statistics & Probability

Interquartile range: $IQR = Q_{3} - Q_{1}$
Mean of a data set: $\bar{x} = \frac{\sum_{i = 1}^{n} f_{i} x_{i}}{n}, n = \sum f_{i}$
Probability of an event: $P (A) = \frac{n (A)}{n (U)}$
Complementary events: $P (A^{'}) = 1 - P (A)$
Combined events: $P (A \cup B) = P (A) + P (B) - P (A \cap B)$
Mutually exclusive events: $P (A \cap B) = 0$
Conditional probability: $P (A ∣ B) = \frac{P (A \cap B)}{P (B)}$
Independent events: $P (A \cap B) = P (A) P (B)$
Expected value (discrete): $E (X) = \sum x P (X = x)$

Topic 5 — Calculus

Derivative of xⁿ: $f (x) = x^{n} ⟹ f^{'} (x) = n x^{n - 1}$
Integral of xⁿ: $\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C, n \neq - 1$
Area between curve and x-axis: $A = \int_{a}^{b} y d x$
Trapezoidal rule (approx. area): $A \approx \frac{h}{2} (y_{0} + y_{n} + 2 (y_{1} + y_{2} + \dots + y_{n - 1})), h = \frac{b - a}{n}$

Not in the AI HL booklet: 46 results to learn

The IB Maths AI HL formula booklet does not print these (SL and HL results), but questions expect you to know them. Learn them by heart; every formula above without a ★ is given in the exam.

Prior learning

Pythagoras; SOHCAHTOA; indices: $a^{2} + b^{2} = c^{2}; \tan θ = \frac{opp}{adj}; a^{m} a^{n} = a^{m + n}$

Number & algebra

Standard form: $a \times 10^{k}, 1 \leq a < 10$
Modulus, argument: $| z | = \sqrt{a^{2} + b^{2}}, \arg z = θ, \tan θ = \frac{b}{a} (check quadrant)$
Products: $| z_{1} z_{2} | = | z_{1} | | z_{2} |, \arg (z_{1} z_{2}) = \arg z_{1} + \arg z_{2}$
Sinusoids: $A \cos (ω t) + B \cos (ω t + ϕ)$ = real part of $(A + B e^{i ϕ}) e^{i ω t}$ , so amplitudes and phases add as complex numbers
3×3 determinant: $det A = a | \begin{matrix} e & f \\ h & i \end{matrix} | - b | \begin{matrix} d & f \\ g & i \end{matrix} | + c | \begin{matrix} d & e \\ g & h \end{matrix} |$
Matrix rules: $A B \neq B A in general, (A B)^{- 1} = B^{- 1} A^{- 1}, A X = B \Rightarrow X = A^{- 1} B$
Eigenvalues; eigenvectors: $det (M - λ I) = 0; M p = λ p$
$P$ = eigenvectors as columns, $D$ = diagonal matrix of the matching eigenvalues

Functions & models

Perpendicular: $m_{1} m_{2} = - 1$
Models: $f (x) = k a^{x} + c, k e^{r x} + c; a x^{n}; a \sin (b x) + d (period \frac{360^{\circ}}{b})$
Logistic model: $f (x) = \frac{L}{1 + C e^{- k x}} (L, C, k > 0)$
Linearising: $y = a x^{b} \Rightarrow \ln y = \ln a + b \ln x; y = k a^{x} \Rightarrow \ln y = \ln k + x \ln a$
Composite; inverse: $(f \circ g) (x) = f (g (x)); f (f^{- 1} (x)) = x$
Transformations: $f (x) + b$ up; $f (x - a)$ right; $p f (x)$ vertical × $p$ ; $f (q x)$ horizontal × $\frac{1}{q}$

Geometry & trigonometry

Radians; bearings: $π rad = 180^{\circ}; bearings clockwise from north$
Perpendicular bisectors & Voronoi: cell edges on perpendicular bisectors of sites; largest empty circle at a vertex or on the boundary
Area of image: $| det A | \times area of object$
Composite transformation: $B$ then $A$ is $A B$
Motion with constant velocity: $r = r_{0} + t v, speed = | v |$
Unit vector; perpendicular: $\hat{v} = \frac{v}{| v |}; v \cdot w = 0$
Component of $v$ along $w$: $v \cdot \hat{w}$
Graphs: sum of degrees = $2 \times$ edges; $K_{n}$ has $\frac{1}{2} n (n - 1)$ edges; a tree on $n$ vertices has $n - 1$ edges
Planar connected graph: $v - e + f = 2, e \leq 3 v - 6 (v \geq 3)$
Eulerian circuit: every vertex even; Eulerian trail: exactly two odd vertices
Walks of length $k$ from $i$ to $j$: $(A^{k})_{i j}$
Chinese postman: add the cheapest pairing of odd vertices, repeated once each
TSP: upper bound = nearest neighbour; lower bound = MST of the rest + two shortest edges from the deleted vertex

Statistics & probability

Outliers: $< Q_{1} - 1.5 I Q R or > Q_{3} + 1.5 I Q R$
Variance: $Var (X) = E (X^{2}) - [E (X)]^{2}$
Binomial pmf: $P (X = r) = (\binom{n}{r}) p^{r} (1 - p)^{n - r}$
Poisson $X \sim Po (m)$: $P (X = x) = \frac{m^{x} e^{- m}}{x!}$
Sum of independent Poissons: $Po (m_{1}) + Po (m_{2}) = Po (m_{1} + m_{2})$
Sample mean (CLT, $n \geq 30$ or $X$ normal): $\bar{X} \sim N (μ, \frac{σ^{2}}{n})$
Linear combinations of independent normals are normal; for $n$ independent $X_{i} \sim N (μ, σ^{2})$ : $\sum X_{i} \sim N (n μ, n σ^{2})$
Expected frequency; df: $\frac{row \times column}{total}; (r - 1) (c - 1)$
Goodness of fit df: classes − 1 − parameters estimated
Reject $H_{0}$ if $p$ -value $<$ significance level. Type I: reject a true $H_{0}$ ; Type II: accept a false $H_{0}$
Transition matrix: $T^{n} s_{0} = s_{n}; steady state T s = s, \sum s_{i} = 1$

Calculus

Tangent; normal: $y - f (a) = f^{'} (a) (x - a); m_{n} = - \frac{1}{f^{'} (a)}$
Max / min; inflexion: $f^{'} = 0, f^{″} < 0 / f^{″} > 0; f^{″} = 0 with sign change$
Related rates: $\frac{d V}{d t} = \frac{d V}{d r} \cdot \frac{d r}{d t}$
Linear inside; by inspection: $\int f (a x + b) d x = \frac{1}{a} F (a x + b) + C; \int g^{'} (x) f^{'} (g (x)) d x = f (g (x)) + C$
Separable: $\frac{d y}{d x} = f (x) g (y) \Rightarrow \int \frac{d y}{g (y)} = \int f (x) d x$
Phase portraits: real $λ$ both $< 0$ stable node, both $> 0$ unstable, opposite signs saddle; complex: $Re λ < 0$ spiral in, $> 0$ out, $= 0$ closed loops
2nd order $\ddot{x} + a \dot{x} + b x = 0$: $put y = \dot{x} : \dot{x} = y, \dot{y} = - b x - a y$

Each of these with a worked example · Tick them off on the interactive sheet · One-page PDF of just these

Using the IB Maths AI HL formula booklet: mistakes that cost marks

Degrees or radians. The AI HL booklet prints the arc and sector formulas in both degrees and radians. Check which unit the question uses before you pick one.
The biased variance. For an unbiased estimate of the population variance use $s_{n - 1}^{2} = \frac{n}{n - 1} s_{n}^{2}$ . With $n = 5$ and $s_{n}^{2} = 4$ , the estimate is $s_{n - 1}^{2} = 5$ .
Matrices in the wrong order. Transformation $B$ followed by transformation $A$ is the matrix $A B$ , not $B A$ ; in general $A B \neq B A$ .
Rounding inside Euler's method. Keep each $y_{n}$ to full accuracy (use the GDC's table or Ans) and round only the final estimate; early rounding drifts over many steps.

Calculator tip: Every AI HL paper allows the GDC. Use Poisson and binomial cdf for probabilities, matrix powers $T^{n}$ for Markov chains and the $χ^{2}$ test for goodness of fit, and write the probability or matrix statement before the number.

IB Maths AI HL formula sheet PDF: download it your way

Our own one-page sheets (not the IB's booklet), free to print, no sign-up.

Preview of our free one-page IB Maths AI HL formula sheet: every formula on one A4 page, colour-coded by topic, with a star on each one the booklet leaves out — The A4 sheet (PDF)

Key formulas explained, with a worked example

Binomial distribution Chain rule Derivative of sin x Product rule Pythagoras' theorem Quotient rule Sum to infinity of a geometric series Trig identities

Questions

Is this the IB's own formula booklet?

No. It is our own page, written in the same notation as the IB formula booklet so you get used to it. In the exam you are given a clean copy of the IB booklet.

Can I take this sheet into the exam?

No. Only the clean copy of the IB formula booklet that the exam provides is allowed. Use this page and the PDF to revise.

Do HL students need the SL formulas too?

Yes. HL papers can use every SL formula as well as these HL ones. The free IB Maths AI HL PDF has both on one page.

Do you get the formula booklet in every IB Maths AI HL paper?

Yes. A clean copy of the formula booklet is provided in all three papers. You cannot bring your own copy or any notes.

How do I get quick at using the formula booklet?

Practise past-style questions with the booklet open beside you, so you know which topic each formula sits under and what each letter means. Then learn the results it leaves out (our formulas-to-memorise page lists them with worked examples).

Is there a separate formula booklet for SL and HL?

Yes. AI has an SL booklet and an HL booklet. The HL booklet includes the SL formulas as well as the HL ones, so HL students need only the HL booklet.

Is the IB Maths AI HL data booklet the same as the formula booklet?

Yes. For maths the IB calls it the formula booklet; "data booklet" is the name of the IB Chemistry and Physics booklets, so many students use either name. Everything on this page is in the IB Maths AI HL formula booklet, apart from the results marked as not in the booklet.

Which formula booklet is used for the 2026 exams?

The current editions, for first assessment 2021. They are used for every exam on the current courses, up to the last ones in November 2028. The revised courses are first examined in May 2029.

Is the PDF free?

Yes. The PDFs (A4, A3 poster, black and white, and the formulas to memorise) and this page are free, with no sign-up.

What else is free here

Make a free account: no card needed5 more practice questions on every topic with instant feedback, full worked answers on free questions, 3 AI-marked answers a week, free, and your streak and progress on any device.

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Other courses: IB Math AI SL formula sheet · IB Math AA SL formula booklet · IB Math AA HL formula booklet · Print-ready one-page booklet · HL AI revision notes · HL AI questions by topic

IB Maths AI HL Formula Booklet — Applications & Interpretation HL

What's in the IB Maths AI HL formula booklet?

Everything in SL AI, plus…

Topic 3 (HL only) — Geometry & Trig extension

Using the booklet: a worked example

Topic 4 (HL only) — Statistics extension

Topic 5 (HL only) — Calculus extension

The SL formulas (also in the HL booklet)

Prior learning

Topic 1 — Number & Algebra

Topic 2 — Functions

Topic 3 — Geometry & Trigonometry

Topic 4 — Statistics & Probability

Topic 5 — Calculus

Try it now

Not in the AI HL booklet: 46 results to learn

Prior learning

Number & algebra

Functions & models

Geometry & trigonometry

Statistics & probability

Calculus

Using the IB Maths AI HL formula booklet: mistakes that cost marks

IB Maths AI HL formula sheet PDF: download it your way

Key formulas explained, with a worked example

Questions

What else is free here