The IB Maths formula booklet, AA and AI
Every IB Mathematics paper comes with a clean copy of your course's formula booklet. Knowing what is in it saves memorising; knowing what is not saves marks. This guide covers both Analysis and Approaches and Applications and Interpretation.
On this page
Two booklets, one for each course
The IB publishes one formula booklet for Mathematics: Analysis and Approaches (AA) and another for Mathematics: Applications and Interpretation (AI). Each covers both SL and HL: formulas for the additional higher level content are marked AHL, and SL students can ignore them. The booklets were introduced with the courses first examined in 2021.
A clean copy is provided in every paper, so you cannot use your own annotated copy. Practise all course with the same version you will see in the exam.
How the booklet is organised
Both booklets start with prior learning: formulas from earlier years, such as areas and volumes of common shapes and coordinate geometry. Then they follow the five syllabus topics:
- Number and algebra
- Functions
- Geometry and trigonometry
- Statistics and probability
- Calculus
The main formulas by topic
This is a study summary, not a copy of the booklet. Check exact forms and AHL labels in your own clean copy.
- Number and algebra: the nth term and sum of arithmetic and geometric sequences, the sum to infinity of a geometric series, compound interest, laws of logarithms and the change of base formula. AA adds the binomial theorem; at AHL there are complex numbers. AI includes more financial mathematics, and matrices at AHL.
- Functions: the equation of a straight line, the axis of symmetry and roots of a quadratic, and, in AI, models such as exponential and logistic functions.
- Geometry and trigonometry: distance and midpoint in three dimensions, volumes and surface areas of pyramids, cones and spheres, the sine rule, the cosine rule and the area of a triangle, and arc length and sector area in radians. AA includes trigonometric identities and double angle formulas; at AHL both courses add vectors.
- Statistics and probability: the interquartile range, mean of grouped data, probability of combined events, conditional probability, expected value, and the mean and variance of the binomial distribution. AI includes more statistical tests, such as the chi-squared test.
- Calculus: standard derivatives such as those of xn, sin x, cos x, ex and ln x, standard integrals, the area under a curve, and in AA the chain, product and quotient rules. At AHL there is more, including volumes of revolution.
How the AA and AI booklets differ
The two booklets share prior learning and the basic formulas, but follow their courses. The AA booklet carries more pure mathematics, such as the binomial theorem, trigonometric identities and, at AHL, further calculus. The AI booklet carries more applied mathematics, such as financial formulas, statistical tests and, at AHL, matrices and models. If you switch course, switch booklets: practising with the wrong one gives you a false picture of what you will be given.
What you still need to know
- When to use each formula: the booklet gives the arithmetic sum formula, but not how to recognise that a problem is arithmetic.
- Exact trigonometric values, such as sin 30° = ½ and cos 45° = √2/2, which you need without a calculator in AA Paper 1.
- Algebraic techniques: factorising, completing the square, solving inequalities and manipulating indices.
- Definitions and conditions, for example that a geometric series converges only when |r| < 1.
- How to use your calculator for statistics, graphs and equations in calculator papers.
Using the booklet: a geometric series
A geometric series has first term 18 and common ratio 2/3. Find its sum to infinity.
- The booklet gives S∞ = u1 / (1 − r) for |r| < 1.
- |2/3| < 1, so the series converges.
- S∞ = 18 / (1/3) = 54.
Combining two formulas
In triangle ABC, a = 7 cm, b = 8 cm and C = 60°. Find side c and the area.
- Cosine rule: c² = 49 + 64 − 2 × 7 × 8 × cos 60° = 113 − 56 = 57, so c = 7.55 cm.
- Area = ½ab sin C = ½ × 7 × 8 × sin 60° = 24.2 cm².
- Both formulas are in the booklet. Knowing that cos 60° = ½ lets you do the first step without a calculator.
How to train with it
- Print a clean copy and use it for every past paper.
- Before revising a topic, read its page so you know what you will be given.
- Make a short list of what is not in the booklet for each topic and learn that list.
- Time yourself finding formulas: in a 90-minute paper, seconds spent searching add up.
Common mistakes
- Using the AI booklet for AA, or the other way round: they are different.
- Using a formula outside its conditions, such as a sum to infinity when |r| ≥ 1.
- Working in degrees when a calculus or arc length formula needs radians.
- Assuming a formula is given because it was given in a different course.
Practice questions
Try each one before opening the answer.
Q1. The 5th term of an arithmetic sequence is 23 and the 12th is 51. Find the first term and common difference.
Show answer
Q2. Find the arc length of a sector with radius 6 cm and angle 1.2 radians.
Show answer
Q3. X ~ B(10, 0.3). Find E(X) and Var(X).
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Frequently asked questions
Is the formula booklet the same for SL and HL?
Each course has one booklet for both levels. Higher level formulas are marked AHL.
Can I bring my own formula sheet?
No. A clean copy of the official booklet is provided in every paper.
Is the derivative of tan x in the booklet?
Check your own copy for your course. Whatever is given, learn the standard derivatives well enough to use them quickly.
Key takeaways
- Separate booklets for AA and AI, each covering SL and HL.
- Prior learning, then the five topics.
- AHL formulas are labelled.
- Exact values, techniques and conditions are not given.
- Train with a clean copy all course.