IB Mathematics · AI SL

IB Math AI SL Formula Booklet — Every Formula, Explained

If you are sitting IB Mathematics: Applications and Interpretation SL, you receive the official AI formula booklet (often searched for as the AI SL formula sheet or data booklet) in Paper 1 and Paper 2. SL students use only the sections without the HL label; every one of them is listed below.

AI SL is a calculator course, so the booklet is short and the real skill is knowing which formula or GDC command the question wants. Learn the list, then the things the booklet leaves 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 SL formula booklet — section by section

Numbered to match the syllabus sub-topics printed in the booklet. Only the SL sections are listed — HL-only entries in the same booklet are not examined at SL.

Prior learning

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

\(\displaystyle \int_a^b y\,dx \approx \tfrac{1}{2}h\big[(y_0 + y_n) + 2(y_1 + y_2 + \cdots + y_{n-1})\big],\quad h = \dfrac{b-a}{n}\)

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 SL and absent from the booklet.

Which hypothesis test to use — \(\chi^2\) goodness of fit, \(\chi^2\) test for independence, or the \(t\)-test — and how to write \(H_0\) and \(H_1\)
Pearson's \(r\) and Spearman's \(r_s\): what they measure and when each is appropriate (computed on the GDC)
Linear regression \(y = ax + b\) and the difference between the \(y\)-on-\(x\) line and interpolation vs extrapolation
Right-angled trigonometry: SOH-CAH-TOA, angles of elevation and depression, bearings measured clockwise from north
Perpendicular bisectors and how Voronoi cells are built from them
Laws of exponents and how to solve \(a^x = b\) with logs on the GDC
The shapes of linear, quadratic, exponential, cubic and sinusoidal models and what each parameter does
Upper and lower bounds, significant figures and rounding rules for accuracy questions
The normal distribution shape, symmetry and the inverse normal on the GDC
Tangents and normals: gradient of the normal is \(-\dfrac{1}{f'(x)}\)

The 10 formulas AI SL 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 — the start of every savings, loan and depreciation question before the TVM solver.
\(\displaystyle \varepsilon = \left|\dfrac{v_A - v_E}{v_E}\right| \times 100\%\)
Percentage error — asked whenever a model or an estimate is compared with a true value.
\(\displaystyle \dfrac{a}{\sin A} = \dfrac{b}{\sin B}\)
Sine rule — non-right-angled triangles in surveying and navigation contexts.
\(\displaystyle c^2 = a^2 + b^2 - 2ab\cos C\)
Cosine rule — two sides and the included angle, or three sides to find an angle.
\(\displaystyle A = \dfrac{\theta}{360}\times\pi r^2\)
Area of a sector (degrees) — AI SL works in degrees, so this is the form you need.
\(\displaystyle V = \tfrac{1}{3}\pi r^2 h\)
Volume of a cone — 3D shapes appear in both papers; the sphere and pyramid are on the same page.
\(\displaystyle u_n = u_1 r^{\,n-1}\)
Geometric sequence — growth and decay questions phrased as sequences.
\(\displaystyle P(A \cup B) = P(A) + P(B) - P(A \cap B)\)
Combined events — Venn-diagram questions almost always need this.
\(\displaystyle E(X) = np,\quad \text{Var}(X) = np(1-p)\)
Binomial distribution — identify \(n\) and \(p\) from the context first.
\(\displaystyle \int_a^b y\,dx \approx \tfrac{1}{2}h\big[(y_0 + y_n) + 2(y_1 + \cdots + y_{n-1})\big]\)
Trapezoidal rule — area from a table of values; count the strips carefully.

How to use the booklet, paper by paper

Paper 1 (calculator, short questions)

Many short questions across every topic. Compound interest, percentage error, sine/cosine rules, sector area and the binomial formulas are the pages you will open most.

Paper 2 (calculator, long questions)

Modelling and statistics in context. The booklet provides little here — the work is choosing the right model and running the GDC correctly. Practise the \(\chi^2\) and \(t\)-test routines and regression until they are automatic.

AI SL formula booklet — common questions

Is the AI SL formula booklet the same as the AI HL one?

Yes. One Applications and Interpretation booklet covers both; SL students use the sections without the HL label. AA students have a different booklet.

Is it a formula sheet, formula booklet or data booklet?

All three names are used for the same document. The IB's name is formula booklet.

Are the \(\chi^2\) and \(t\)-test formulas included?

No. These tests are run on the GDC; the booklet gives only basic probability and the binomial mean and variance.

Does AI SL use radians?

No — AI SL works in degrees, which is why the arc-length and sector formulas are written with \(\tfrac{\theta}{360}\). Radians appear only at HL.

Where do I get the official booklet?

From your IB coordinator; it is published by the IB in the Programme Resource Centre. Use the version your school issues.

Want the booklet to feel like second nature?

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