IB Mathematics · AI HL
IB Math AI HL Formula Booklet — Every Formula, Explained
If you are sitting IB Mathematics: Applications and Interpretation HL, you receive the official AI formula booklet (the data booklet) in Paper 1, Paper 2 and Paper 3. It is shared with AI SL — the HL material (matrices, vectors, Poisson, Euler's method, coupled systems) is marked HL. Everything in it is listed below, topic by topic.
AI HL is a calculator-heavy course, so the booklet is less about algebra and more about knowing which model or test applies. Use this page to learn where each formula lives, then the list of what you must memorise because the booklet leaves it out.
Check against your own copy. This page follows the IB Applications and Interpretation formula booklet for first assessment 2021 onwards. The IB occasionally reissues the booklet with small layout changes — always use the version your school gives you, and tell Mr Ahmad if you spot a difference.
What is in the AI HL formula booklet — section by section
Numbered to match the syllabus sub-topics printed in the booklet. Sections marked HL are the additions HL students use on top of the SL core.
Prior learning (SL and HL)
Area of a parallelogram
\(\displaystyle A = bh\)
Area of a triangle
\(\displaystyle A = \tfrac{1}{2}bh\)
Area of a trapezoid
\(\displaystyle A = \tfrac{1}{2}(a+b)h\)
Area and circumference of a circle
\(\displaystyle A = \pi r^2,\quad C = 2\pi r\)
Volume of a cuboid
\(\displaystyle V = lwh\)
Volume of a cylinder
\(\displaystyle V = \pi r^2 h\)
Volume of a prism
\(\displaystyle V = Ah\)
Curved surface area of a cylinder
\(\displaystyle A = 2\pi r h\)
Distance between two points
\(\displaystyle d = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}\)
Midpoint
\(\displaystyle \left(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2}\right)\)
Topic 1 — Number and algebra
1.2 Arithmetic sequence, nth term
\(\displaystyle u_n = u_1 + (n-1)d\)
1.2 Sum of an arithmetic sequence
\(\displaystyle S_n = \tfrac{n}{2}\big(2u_1 + (n-1)d\big) = \tfrac{n}{2}(u_1 + u_n)\)
1.3 Geometric sequence, nth term
\(\displaystyle u_n = u_1 r^{\,n-1}\)
1.3 Sum of a geometric sequence
\(\displaystyle S_n = \dfrac{u_1(r^n - 1)}{r-1} = \dfrac{u_1(1-r^n)}{1-r},\quad r \ne 1\)
1.4 Compound interest
\(\displaystyle FV = PV\left(1 + \dfrac{r}{100k}\right)^{kn}\)
1.5 Exponents and logarithms
\(\displaystyle a^x = b \iff x = \log_a b,\quad a>0,\ b>0,\ a\ne 1\)
1.6 Percentage error
\(\displaystyle \varepsilon = \left|\dfrac{v_A - v_E}{v_E}\right| \times 100\%\)
Topic 2 — Functions
2.1 Equations of a straight line
\(\displaystyle y = mx + c;\quad ax + by + d = 0;\quad y - y_1 = m(x - x_1)\)
2.1 Gradient
\(\displaystyle m = \dfrac{y_2 - y_1}{x_2 - x_1}\)
2.5 Axis of symmetry of a quadratic
\(\displaystyle f(x) = ax^2 + bx + c \ \Rightarrow\ x = -\dfrac{b}{2a}\)
Topic 3 — Geometry and trigonometry
3.1 Distance between two points in 3D
\(\displaystyle d = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2 + (z_1-z_2)^2}\)
3.1 Midpoint in 3D
\(\displaystyle \left(\dfrac{x_1+x_2}{2},\ \dfrac{y_1+y_2}{2},\ \dfrac{z_1+z_2}{2}\right)\)
3.1 Volume of a right pyramid
\(\displaystyle V = \tfrac{1}{3}Ah\)
3.1 Volume and curved surface of a cone
\(\displaystyle V = \tfrac{1}{3}\pi r^2 h,\qquad A = \pi r l\)
3.1 Sphere
\(\displaystyle V = \tfrac{4}{3}\pi r^3,\qquad A = 4\pi r^2\)
3.2 Sine rule
\(\displaystyle \dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}\)
3.2 Cosine rule
\(\displaystyle c^2 = a^2 + b^2 - 2ab\cos C;\qquad \cos C = \dfrac{a^2 + b^2 - c^2}{2ab}\)
3.2 Area of a triangle
\(\displaystyle A = \tfrac{1}{2}ab\sin C\)
3.4 Length of an arc (degrees)
\(\displaystyle l = \dfrac{\theta}{360}\times 2\pi r\)
3.4 Area of a sector (degrees)
\(\displaystyle A = \dfrac{\theta}{360}\times \pi r^2\)
Topic 4 — Statistics and probability
4.2 Interquartile range
\(\displaystyle IQR = Q_3 - Q_1\)
4.3 Mean of a data set
\(\displaystyle \bar{x} = \dfrac{\sum_{i=1}^{k} f_i x_i}{n},\quad n = \sum_{i=1}^{k} f_i\)
4.5 Probability of an event
\(\displaystyle P(A) = \dfrac{n(A)}{n(U)}\)
4.5 Complementary events
\(\displaystyle P(A) + P(A') = 1\)
4.6 Combined events
\(\displaystyle P(A \cup B) = P(A) + P(B) - P(A \cap B)\)
4.6 Mutually exclusive events
\(\displaystyle P(A \cup B) = P(A) + P(B)\)
4.6 Conditional probability
\(\displaystyle P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}\)
4.6 Independent events
\(\displaystyle P(A \cap B) = P(A)\,P(B)\)
4.7 Expected value of a discrete random variable
\(\displaystyle E(X) = \sum x\,P(X = x)\)
4.8 Binomial distribution
\(\displaystyle X \sim B(n,p):\quad E(X) = np,\qquad \text{Var}(X) = np(1-p)\)
Topic 5 — Calculus
5.3 Derivative of a power
\(\displaystyle f(x) = ax^n \ \Rightarrow\ f'(x) = anx^{n-1}\)
5.5 Integral of a power
\(\displaystyle \int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C,\quad n \ne -1\)
5.5 Area between a curve and the x-axis
\(\displaystyle A = \int_a^b y\,dx\)
5.8 Trapezoidal rule
\(\displaystyle \int_a^b y\,dx \approx \tfrac{1}{2}h\big[(y_0 + y_n) + 2(y_1 + y_2 + \cdots + y_{n-1})\big],\quad h = \dfrac{b-a}{n}\)
Topic 1 — Number and algebra HL
1.9 Laws of logarithms
\(\displaystyle \log_a xy = \log_a x + \log_a y;\quad \log_a \dfrac{x}{y} = \log_a x - \log_a y;\quad \log_a x^m = m\log_a x\)
1.10 Sum of an infinite geometric sequence
\(\displaystyle S_\infty = \dfrac{u_1}{1-r},\quad |r|<1\)
1.12 Complex numbers
\(\displaystyle z = a + bi\)
1.13 Modulus–argument (polar) and Euler form
\(\displaystyle z = r(\cos\theta + i\sin\theta) = re^{i\theta} = r\,\text{cis}\,\theta\)
1.14 Determinant and inverse of a 2×2 matrix
\(\displaystyle \mathbf{A} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \Rightarrow \det\mathbf{A} = ad - bc,\quad \mathbf{A}^{-1} = \dfrac{1}{\det\mathbf{A}}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix},\ ad \ne bc\)
1.15 Power of a diagonalised matrix
\(\displaystyle \mathbf{M}^n = \mathbf{P}\mathbf{D}^n\mathbf{P}^{-1}\)
Topic 2 — Functions HL
2.9 Logistic function
\(\displaystyle f(x) = \dfrac{L}{1 + Ce^{-kx}},\quad L, C, k > 0\)
Topic 3 — Geometry and trigonometry HL
3.7 Arc length and sector area (radians)
\(\displaystyle l = r\theta;\qquad A = \tfrac{1}{2}r^2\theta\)
3.8 Trigonometric identities
\(\displaystyle \tan\theta = \dfrac{\sin\theta}{\cos\theta};\qquad \cos^2\theta + \sin^2\theta = 1\)
3.9 Transformation matrices
\(\displaystyle \text{Reflection in } y = (\tan\theta)x: \begin{pmatrix} \cos 2\theta & \sin 2\theta \\ \sin 2\theta & -\cos 2\theta \end{pmatrix};\quad \text{Rotation by } \theta \text{ anticlockwise}: \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}\)
3.10 Magnitude of a vector
\(\displaystyle |\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}\)
3.13 Scalar product and angle between vectors
\(\displaystyle \mathbf{v}\cdot\mathbf{w} = v_1w_1 + v_2w_2 + v_3w_3 = |\mathbf{v}||\mathbf{w}|\cos\theta\)
3.14 Vector equation of a line
\(\displaystyle \mathbf{r} = \mathbf{a} + \lambda\mathbf{b}\)
3.16 Vector product
\(\displaystyle \mathbf{v}\times\mathbf{w} = \begin{pmatrix} v_2w_3 - v_3w_2 \\ v_3w_1 - v_1w_3 \\ v_1w_2 - v_2w_1 \end{pmatrix};\qquad |\mathbf{v}\times\mathbf{w}| = |\mathbf{v}||\mathbf{w}|\sin\theta\)
3.16 Area of a parallelogram (vectors)
\(\displaystyle A = |\mathbf{v}\times\mathbf{w}|\)
Topic 4 — Statistics and probability HL
4.14 Linear transformation of a random variable
\(\displaystyle E(aX + b) = aE(X) + b;\qquad \text{Var}(aX + b) = a^2\,\text{Var}(X)\)
4.14 Linear combinations of independent random variables
\(\displaystyle E(a_1X_1 \pm a_2X_2) = a_1E(X_1) \pm a_2E(X_2);\qquad \text{Var}(a_1X_1 \pm a_2X_2) = a_1^2\text{Var}(X_1) + a_2^2\text{Var}(X_2)\)
4.14 Sample mean
\(\displaystyle E(\bar{X}) = \mu;\qquad \text{Var}(\bar{X}) = \dfrac{\sigma^2}{n}\)
4.14 Unbiased estimate of population variance
\(\displaystyle s_{n-1}^2 = \dfrac{n}{n-1}\,s_n^2\)
4.17 Poisson distribution
\(\displaystyle X \sim \text{Po}(m):\quad E(X) = m,\qquad \text{Var}(X) = m\)
Topic 5 — Calculus HL
5.9 Standard derivatives
\(\displaystyle \dfrac{d}{dx}\sin x = \cos x;\quad \dfrac{d}{dx}\cos x = -\sin x;\quad \dfrac{d}{dx}e^x = e^x;\quad \dfrac{d}{dx}\ln x = \dfrac{1}{x}\)
5.9 Chain, product and quotient rules
\(\displaystyle \dfrac{dy}{dx} = \dfrac{dy}{du}\cdot\dfrac{du}{dx};\quad (uv)' = u\dfrac{dv}{dx} + v\dfrac{du}{dx};\quad \left(\dfrac{u}{v}\right)' = \dfrac{v\dfrac{du}{dx} - u\dfrac{dv}{dx}}{v^2}\)
5.11 Standard integrals
\(\displaystyle \int \dfrac{1}{x}\,dx = \ln|x| + C;\quad \int \sin x\,dx = -\cos x + C;\quad \int \cos x\,dx = \sin x + C;\quad \int e^x\,dx = e^x + C\)
5.12 Area and volume of revolution
\(\displaystyle A = \int_a^b |y|\,dx;\qquad V = \int_a^b \pi y^2\,dx\)
5.13 Kinematics
\(\displaystyle v = \dfrac{ds}{dt};\quad a = \dfrac{dv}{dt} = \dfrac{d^2s}{dt^2};\quad \text{distance} = \int_{t_1}^{t_2}|v|\,dt;\quad \text{displacement} = \int_{t_1}^{t_2} v\,dt\)
5.16 Euler's method
\(\displaystyle y_{n+1} = y_n + h\,f(x_n, y_n);\qquad x_{n+1} = x_n + h\)
5.16 Euler's method for coupled systems
\(\displaystyle x_{n+1} = x_n + h\,f_1(x_n, y_n, t_n);\quad y_{n+1} = y_n + h\,f_2(x_n, y_n, t_n);\quad t_{n+1} = t_n + h\)
5.17 Exact solution of a coupled linear system
\(\displaystyle \mathbf{x} = A e^{\lambda_1 t}\mathbf{p}_1 + B e^{\lambda_2 t}\mathbf{p}_2\)
Not in the booklet — you must memorise these
Students lose marks every session looking for things that were never there. Everything below is examinable in AI HL and absent from the booklet.
- Which hypothesis test to use — \(\chi^2\) goodness of fit, \(\chi^2\) independence, \(t\)-test, binomial or Poisson tests — and how to state \(H_0\) and \(H_1\)
- The meaning of Type I and Type II errors and how the significance level relates to the critical region
- Pearson's \(r\) and Spearman's \(r_s\): when each is appropriate (the formulas are not in the booklet — use the GDC)
- How to linearise data: \(y = ax^n \Rightarrow \ln y = \ln a + n\ln x\); \(y = ab^x \Rightarrow \ln y = \ln a + x\ln b\)
- Graph theory definitions: Eulerian and Hamiltonian, trails, circuits, trees, and how Kruskal's, Prim's and Dijkstra's algorithms run
- Matrix multiplication, identity and zero matrices, and the conditions for an inverse (\(\det \ne 0\))
- Eigenvalues from \(\det(\mathbf{A} - \lambda\mathbf{I}) = 0\) and eigenvectors from \((\mathbf{A} - \lambda\mathbf{I})\mathbf{x} = \mathbf{0}\)
- Markov chains: transition matrices, steady state from \(\boldsymbol{\pi}\mathbf{P} = \boldsymbol{\pi}\)
- Phase portraits: how the signs of eigenvalues decide stable, unstable and saddle points
- Voronoi diagrams: perpendicular bisectors, cells, and the largest empty circle problem
The 10 formulas AI HL students use most
From marking hundreds of practice papers: these come up in almost every exam. Know their page in the booklet and what each symbol means.
- \(\displaystyle FV = PV\left(1 + \dfrac{r}{100k}\right)^{kn}\)Compound interest — also the backbone of loan and annuity questions, which you then finish on the TVM solver.
- \(\displaystyle \varepsilon = \left|\dfrac{v_A - v_E}{v_E}\right| \times 100\%\)Percentage error — appears in almost every modelling question that compares a model with data.
- \(\displaystyle c^2 = a^2 + b^2 - 2ab\cos C\)Cosine rule — surveying, bearings and 3D geometry questions.
- \(\displaystyle \mathbf{A}^{-1} = \dfrac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}\)Inverse of a 2×2 matrix — solving systems, decoding, and finding \(\mathbf{P}^{-1}\) for diagonalisation.
- \(\displaystyle \mathbf{M}^n = \mathbf{P}\mathbf{D}^n\mathbf{P}^{-1}\)Powers of a matrix via diagonalisation — long-run Markov chains and population models.
- \(\displaystyle f(x) = \dfrac{L}{1 + Ce^{-kx}}\)Logistic model — any growth with a ceiling: populations, adoption, spread of information.
- \(\displaystyle \text{Var}(\bar{X}) = \dfrac{\sigma^2}{n}\)Distribution of the sample mean — underpins confidence intervals and \(t\)-tests on means.
- \(\displaystyle X \sim \text{Po}(m): E(X) = \text{Var}(X) = m\)Poisson distribution — rare events per interval; check that mean ≈ variance before using it.
- \(\displaystyle \int_a^b y\,dx \approx \tfrac{1}{2}h\big[(y_0 + y_n) + 2(y_1 + \cdots + y_{n-1})\big]\)Trapezoidal rule — numerical area when only data points are known.
- \(\displaystyle y_{n+1} = y_n + h\,f(x_n, y_n)\)Euler's method — the standard Paper 3 technique for differential equations without closed-form solutions.
How to use the booklet, paper by paper
Paper 1 (calculator, short questions)
Fast questions across every topic. Compound interest, percentage error, sine and cosine rules, the binomial mean/variance and the trapezoidal rule come up constantly — know their pages.
Paper 2 (calculator, long questions)
Modelling and statistics in depth. The booklet gives you the logistic function, linear-transformation rules for random variables and the sample-mean variance; the hypothesis-test procedure is yours to know.
Paper 3 (extended response)
Two long investigations, often one on matrices or graph theory and one on differential equations. The matrix inverse, diagonalisation power formula, Euler's method and the coupled-system solution are all in the booklet — practise finding them quickly.
AI HL formula booklet — common questions
Is the AI HL formula booklet the same as the AI SL one?
Yes. One Applications and Interpretation booklet covers both levels; HL sections are marked HL. The AA courses use a different booklet.
Formula booklet or data booklet?
Same thing. The IB says formula booklet; many schools say data booklet.
Are the hypothesis-test formulas in the booklet?
No. The \(\chi^2\) and \(t\)-tests are done on the GDC; the booklet only gives distribution means and variances and the sample-mean results.
Are Dijkstra, Prim and Kruskal in the booklet?
No. Algorithms are not listed — you must know how to run them and how to show your working in a table.
Where do I get the official booklet?
From your IB coordinator; the IB publishes it in the Programme Resource Centre. Use the version your school issues.
Want the booklet to feel like second nature?
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