IB Mathematics · AA HL

IB Math AA HL Formula Booklet — Every Formula, Explained

Q: Is the AA HL formula booklet different from the AA SL one?

No. There is one Analysis and Approaches booklet for both levels. HL students use the SL sections plus the sections marked HL. The AI courses use a separate booklet.

Q: Is it called a formula booklet or a data booklet?

The IB calls it the formula booklet. Many schools and students say data booklet. They are the same document.

Q: Can I bring my own annotated copy into the exam?

No. A clean copy is provided by the IB with each paper. Annotate a practice copy at home so you know where everything is.

Q: Does the booklet include the statistical tables?

No tables are needed — the normal, binomial and other distributions are computed on your GDC in Papers 2 and 3.

Q: Where can I download the official booklet?

Ask your school's IB coordinator — the current version is published by the IB in the Programme Resource Centre. Use the version your school issues, since minor layout updates happen between sessions.

If you are sitting IB Mathematics: Analysis and Approaches HL, you walk into Paper 1, Paper 2 and Paper 3 with the official AA formula booklet (often called the data booklet). It is the same booklet AA SL students receive — the HL formulas are simply extra sections marked HL. This page lists every formula in it, topic by topic, so you can learn where each one lives before exam day.

The booklet gives you formulas; it does not tell you when to use them, and it leaves out a surprising amount that examiners expect you to know by heart. Both lists are below.

Check against your own copy. This page follows the IB Analysis and Approaches 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 AA 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

\(\displaystyle a^x = b \iff x = \log_a b,\quad a>0,\ b>0,\ a\ne 1\)

1.7 Laws of logarithms

\(\displaystyle \log_a xy = \log_a x + \log_a y,\quad \log_a \dfrac{x}{y} = \log_a x - \log_a y,\quad \log_a x^m = m\log_a x\)

1.7 Change of base

\(\displaystyle \log_a x = \dfrac{\log_b x}{\log_b a}\)

1.8 Sum of an infinite geometric sequence

\(\displaystyle S_\infty = \dfrac{u_1}{1-r},\quad |r|<1\)

1.9 Binomial theorem

\(\displaystyle (a+b)^n = a^n + \binom{n}{1}a^{n-1}b + \cdots + \binom{n}{r}a^{n-r}b^r + \cdots + b^n\)

1.9 Binomial coefficient

\(\displaystyle \binom{n}{r} = \dfrac{n!}{r!\,(n-r)!}\)

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

\(\displaystyle m = \dfrac{y_2 - y_1}{x_2 - x_1}\)

2.6 Axis of symmetry of a quadratic

\(\displaystyle f(x) = ax^2 + bx + c \ \Rightarrow\ x = -\dfrac{b}{2a}\)

2.7 Solutions of a quadratic equation

\(\displaystyle x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a},\quad a \ne 0\)

2.7 Discriminant

\(\displaystyle \Delta = b^2 - 4ac\)

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 (radians)

\(\displaystyle l = r\theta\)

3.4 Area of a sector (radians)

\(\displaystyle A = \tfrac{1}{2}r^2\theta\)

3.5 Identity for tan

\(\displaystyle \tan\theta = \dfrac{\sin\theta}{\cos\theta}\)

3.6 Pythagorean identity

\(\displaystyle \cos^2\theta + \sin^2\theta = 1\)

3.6 Double angle identities

\(\displaystyle \sin 2\theta = 2\sin\theta\cos\theta;\quad \cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta\)

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

\(\displaystyle X \sim B(n,p):\quad E(X) = np,\qquad \text{Var}(X) = np(1-p)\)

4.12 Standardised normal variable

\(\displaystyle z = \dfrac{x - \mu}{\sigma}\)

Topic 5 — Calculus

5.3 Derivative of a power

\(\displaystyle f(x) = x^n \ \Rightarrow\ f'(x) = nx^{n-1}\)

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

\(\displaystyle A = \int_a^b y\,dx\)

5.6 Standard derivatives

\(\displaystyle \dfrac{d}{dx}\sin x = \cos x;\quad \dfrac{d}{dx}\cos x = -\sin x;\quad \dfrac{d}{dx}e^x = e^x;\quad \dfrac{d}{dx}\ln x = \dfrac{1}{x}\)

5.6 Chain rule

\(\displaystyle y = g(u),\ u = f(x) \ \Rightarrow\ \dfrac{dy}{dx} = \dfrac{dy}{du}\cdot\dfrac{du}{dx}\)

5.6 Product rule

\(\displaystyle y = uv \ \Rightarrow\ \dfrac{dy}{dx} = u\dfrac{dv}{dx} + v\dfrac{du}{dx}\)

5.6 Quotient rule

\(\displaystyle y = \dfrac{u}{v} \ \Rightarrow\ \dfrac{dy}{dx} = \dfrac{v\dfrac{du}{dx} - u\dfrac{dv}{dx}}{v^2}\)

5.9 Kinematics

\(\displaystyle v = \dfrac{ds}{dt};\quad a = \dfrac{dv}{dt} = \dfrac{d^2s}{dt^2};\quad \text{distance} = \int_{t_1}^{t_2}|v|\,dt;\quad \text{displacement} = \int_{t_1}^{t_2} v\,dt\)

5.10 Standard integrals

\(\displaystyle \int \dfrac{1}{x}\,dx = \ln|x| + C;\quad \int \sin x\,dx = -\cos x + C;\quad \int \cos x\,dx = \sin x + C;\quad \int e^x\,dx = e^x + C\)

5.11 Area between curve and x-axis (general)

\(\displaystyle A = \int_a^b |y|\,dx\)

Topic 1 — Number and algebra HL

1.10 Permutations and combinations

\(\displaystyle {}^nP_r = \dfrac{n!}{(n-r)!},\qquad \binom{n}{r} = \dfrac{n!}{r!\,(n-r)!}\)

1.10 Extension of the binomial theorem (n a rational number)

\(\displaystyle (a+b)^n = a^n\left(1 + n\dfrac{b}{a} + \dfrac{n(n-1)}{2!}\left(\dfrac{b}{a}\right)^2 + \cdots\right)\)

1.12 Complex numbers

\(\displaystyle z = a + bi\)

1.13 Modulus–argument (polar) and Euler form

\(\displaystyle z = r(\cos\theta + i\sin\theta) = re^{i\theta} = r\,\text{cis}\,\theta\)

1.13 De Moivre's theorem

\(\displaystyle \big[r(\cos\theta + i\sin\theta)\big]^n = r^n(\cos n\theta + i\sin n\theta) = r^n e^{in\theta} = r^n\,\text{cis}\,n\theta\)

Topic 2 — Functions HL

2.12 Sum and product of the roots of a polynomial

\(\displaystyle a_nx^n + a_{n-1}x^{n-1} + \cdots + a_0 = 0:\quad \sum \text{roots} = -\dfrac{a_{n-1}}{a_n},\qquad \prod \text{roots} = \dfrac{(-1)^n a_0}{a_n}\)

Topic 3 — Geometry and trigonometry HL

3.9 Reciprocal trigonometric identities

\(\displaystyle \sec\theta = \dfrac{1}{\cos\theta};\quad \text{cosec}\,\theta = \dfrac{1}{\sin\theta}\)

3.9 Pythagorean identities

\(\displaystyle 1 + \tan^2\theta = \sec^2\theta;\qquad 1 + \cot^2\theta = \text{cosec}^2\theta\)

3.10 Compound angle identities

\(\displaystyle \sin(A \pm B) = \sin A\cos B \pm \cos A\sin B;\quad \cos(A \pm B) = \cos A\cos B \mp \sin A\sin B;\quad \tan(A \pm B) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A\tan B}\)

3.10 Double angle identity for tan

\(\displaystyle \tan 2\theta = \dfrac{2\tan\theta}{1 - \tan^2\theta}\)

3.12 Magnitude of a vector

\(\displaystyle |\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2}\)

3.13 Scalar product

\(\displaystyle \mathbf{v}\cdot\mathbf{w} = v_1w_1 + v_2w_2 + v_3w_3 = |\mathbf{v}||\mathbf{w}|\cos\theta\)

3.13 Angle between two vectors

\(\displaystyle \cos\theta = \dfrac{v_1w_1 + v_2w_2 + v_3w_3}{|\mathbf{v}||\mathbf{w}|}\)

3.14 Vector equation of a line

\(\displaystyle \mathbf{r} = \mathbf{a} + \lambda\mathbf{b}\)

3.14 Parametric and Cartesian forms of a line

\(\displaystyle x = x_0 + \lambda l,\ y = y_0 + \lambda m,\ z = z_0 + \lambda n;\qquad \dfrac{x - x_0}{l} = \dfrac{y - y_0}{m} = \dfrac{z - z_0}{n}\)

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)

\(\displaystyle A = |\mathbf{v}\times\mathbf{w}|\)

3.17 Vector equation of a plane

\(\displaystyle \mathbf{r} = \mathbf{a} + \lambda\mathbf{b} + \mu\mathbf{c}\)

3.17 Normal and Cartesian forms of a plane

\(\displaystyle \mathbf{r}\cdot\mathbf{n} = \mathbf{a}\cdot\mathbf{n};\qquad ax + by + cz = d\)

Topic 4 — Statistics and probability HL

4.13 Bayes' theorem

\(\displaystyle P(B \mid A) = \dfrac{P(B)\,P(A \mid B)}{P(B)\,P(A \mid B) + P(B')\,P(A \mid B')}\)

4.13 Bayes' theorem for three events

\(\displaystyle P(B_i \mid A) = \dfrac{P(B_i)\,P(A \mid B_i)}{P(B_1)P(A \mid B_1) + P(B_2)P(A \mid B_2) + P(B_3)P(A \mid B_3)}\)

4.14 Variance of a discrete random variable

\(\displaystyle \text{Var}(X) = E(X^2) - \big[E(X)\big]^2 = \sum (x-\mu)^2 P(X=x)\)

4.14 Continuous random variables

\(\displaystyle E(X) = \int_{-\infty}^{\infty} x f(x)\,dx;\qquad \text{Var}(X) = \int_{-\infty}^{\infty} (x-\mu)^2 f(x)\,dx\)

Topic 5 — Calculus HL

5.12 Further standard derivatives

\(\displaystyle \dfrac{d}{dx}\tan x = \sec^2 x;\ \dfrac{d}{dx}\sec x = \sec x\tan x;\ \dfrac{d}{dx}\text{cosec}\,x = -\text{cosec}\,x\cot x;\ \dfrac{d}{dx}\cot x = -\text{cosec}^2 x\)

5.12 Derivatives of a^x and log_a x

\(\displaystyle \dfrac{d}{dx}a^x = a^x\ln a;\qquad \dfrac{d}{dx}\log_a x = \dfrac{1}{x\ln a}\)

5.12 Derivatives of inverse trig functions

\(\displaystyle \dfrac{d}{dx}\arcsin x = \dfrac{1}{\sqrt{1-x^2}};\quad \dfrac{d}{dx}\arccos x = -\dfrac{1}{\sqrt{1-x^2}};\quad \dfrac{d}{dx}\arctan x = \dfrac{1}{1+x^2}\)

5.15 Further standard integrals

\(\displaystyle \int a^x\,dx = \dfrac{a^x}{\ln a} + C;\quad \int \dfrac{1}{a^2 + x^2}\,dx = \dfrac{1}{a}\arctan\dfrac{x}{a} + C;\quad \int \dfrac{1}{\sqrt{a^2 - x^2}}\,dx = \arcsin\dfrac{x}{a} + C,\ |x|<a\)

5.16 Integration by parts

\(\displaystyle \int u\,\dfrac{dv}{dx}\,dx = uv - \int v\,\dfrac{du}{dx}\,dx\)

5.17 Area between a curve and the y-axis

\(\displaystyle A = \int_a^b |x|\,dy\)

5.17 Volumes of revolution

\(\displaystyle V = \int_a^b \pi y^2\,dx \ (\text{about the } x\text{-axis});\qquad V = \int_a^b \pi x^2\,dy \ (\text{about the } y\text{-axis})\)

5.18 Euler's method

\(\displaystyle y_{n+1} = y_n + h\,f(x_n, y_n);\qquad x_{n+1} = x_n + h\)

5.18 Integrating factor

\(\displaystyle \dfrac{dy}{dx} + P(x)y = Q(x) \ \Rightarrow\ \text{integrating factor } e^{\int P(x)\,dx}\)

5.19 Maclaurin series

\(\displaystyle f(x) = f(0) + x f'(0) + \dfrac{x^2}{2!}f''(0) + \cdots\)

5.19 Standard Maclaurin series

\(\displaystyle e^x = 1 + x + \dfrac{x^2}{2!} + \cdots;\ \ln(1+x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \cdots;\ \sin x = x - \dfrac{x^3}{3!} + \cdots;\ \cos x = 1 - \dfrac{x^2}{2!} + \cdots;\ \arctan x = x - \dfrac{x^3}{3} + \cdots\)

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 AA HL and absent from the booklet.

Derivative from first principles: \(f'(x) = \lim_{h\to 0}\dfrac{f(x+h)-f(x)}{h}\)
Exact trigonometric values for \(0, \tfrac{\pi}{6}, \tfrac{\pi}{4}, \tfrac{\pi}{3}, \tfrac{\pi}{2}\) and the unit-circle signs in each quadrant
Transformations of graphs: \(y = f(x) + k\), \(y = f(x - h)\), \(y = pf(x)\), \(y = f(qx)\) and their order
Laws of exponents, factorising, completing the square, and the shapes of \(y = \dfrac{1}{x}\), \(y = e^x\), \(y = \ln x\)
Properties of \(\ln\) and \(e\): \(e^{\ln x} = x\), \(\ln e^x = x\), \(\ln 1 = 0\)
Proof techniques: direct, by contradiction, by counterexample and by induction — the structure of an induction proof is never given
Complex numbers: conjugate, \(|z|^2 = z\bar{z}\), roots occur in conjugate pairs for real polynomials
L'Hôpital's rule, the squeeze theorem idea, and when a limit is of the form \(\tfrac{0}{0}\) or \(\tfrac{\infty}{\infty}\)
How to recognise separable and homogeneous differential equations and the substitution \(y = vx\)
The normal distribution shape, 68–95–99.7 rule, and how to use the inverse normal on your GDC

The 10 formulas AA 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.

\(\displaystyle \dfrac{a}{\sin A} = \dfrac{b}{\sin B}\)
Sine rule — any triangle with a known angle and its opposite side. Watch for the ambiguous case when finding an angle.
\(\displaystyle c^2 = a^2 + b^2 - 2ab\cos C\)
Cosine rule — two sides and the included angle, or three sides to find an angle.
\(\displaystyle \binom{n}{r}a^{n-r}b^r\)
General term of a binomial expansion — most Paper 1 'find the coefficient of \(x^k\)' questions come straight from this.
\(\displaystyle S_\infty = \dfrac{u_1}{1-r}\)
Infinite geometric sum — remember to state \(|r|<1\) before using it.
\(\displaystyle \int u\,\dfrac{dv}{dx}\,dx = uv - \int v\,\dfrac{du}{dx}\,dx\)
Integration by parts — choose \(u\) as the part that simplifies when differentiated (ln, inverse trig, polynomial).
\(\displaystyle V = \int_a^b \pi y^2\,dx\)
Volume of revolution about the \(x\)-axis — square \(y\) first, then integrate; the \(\pi\) stays outside.
\(\displaystyle P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}\)
Conditional probability — the denominator is the condition, not the event you are asked about.
\(\displaystyle z = \dfrac{x-\mu}{\sigma}\)
Standardising a normal variable — essential when \(\mu\) or \(\sigma\) is unknown and you must solve for it.
\(\displaystyle \mathbf{v}\cdot\mathbf{w} = |\mathbf{v}||\mathbf{w}|\cos\theta\)
Scalar product — angle between lines, perpendicularity (\(\mathbf{v}\cdot\mathbf{w} = 0\)) and projections.
\(\displaystyle f(x) = f(0) + x f'(0) + \dfrac{x^2}{2!}f''(0) + \cdots\)
Maclaurin series — Paper 3 loves asking you to build the series of a product or composite from the standard ones.

How to use the booklet, paper by paper

Paper 1 (no calculator)

Expect the booklet to carry the algebra: binomial coefficients, log laws, trig identities, standard derivatives and integrals. Know the location of the double-angle identities and the standard integrals page — you will open it several times.

Paper 2 (calculator)

Statistics and modelling dominate. The binomial mean/variance, standardised normal, conditional probability and Bayes' theorem are all in the booklet; the GDC commands are not — practise them until they are automatic.

Paper 3 (extended response)

Long investigations built on HL calculus, series or vectors. Keep the Maclaurin series, integration by parts and vector product pages tabbed in your practice booklet; in the exam, know their page numbers.

AA HL formula booklet — common questions

Is the AA HL formula booklet different from the AA SL one?