IB Maths AA HL Formula Booklet — Analysis & Approaches HL

Quick answer: yes, you get a clean copy of the IB Maths AA HL formula booklet in every paper: calculator-free Paper 1, and Papers 2 and 3. 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 42 results it leaves out.

The 25 formulas that HL adds for IB Mathematics Analysis & Approaches 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 AA 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 AA HL formula booklet?

There is a separate SL booklet and HL booklet for AA. This table is for the AA 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 AA HL formula booklet by topic: printed, and ★ not printed
TopicPrinted in the booklet★ Not printed: learn these
Prior learningAreas · Circumference; cylinder · Cuboid; prism · Distance; midpointIndices
Number & algebraArithmetic · Geometric · Compound interest · Exponents & logs · Log laws · Change of base · and 5 more\((a+bx)^n=a^n\left(1+\tfrac bax\right)^n\), valid for \(|x|<\left|\tfrac ab\right|\) · Partial fractions · Modulus, argument, conjugate · Products · \(n\)th roots of \(re^{i\theta}\) · For \(|z|=1\) · and 3 more
FunctionsLines · Quadratics · Sum, product of roots of \(\sum_0^na_rx^r\)Perpendicular · \(\Delta>0\) · Factor & remainder theorems · Odd; even · Composite; inverse · Transformations · and 2 more
Geometry & trigonometry3D distance (\(\Delta x=x_1-x_2\), …) · 3D midpoint · Pyramid; cone; sphere · Sine, cosine rules; area · Arc; sector (radians) · Identities · and 11 moreExact values \(0,\frac\pi6,\frac\pi4,\frac\pi3,\frac\pi2\) · Ranges · \(a\sin(b(x-c))+d\), \(b>0\) · Perpendicular · Area of triangle · Angle line–plane; plane–plane · and 2 more
Statistics & probabilityIQR; mean · Probability · Combined events · Independent · Conditional · Bayes · and 6 moreOutliers · Variance · Binomial pmf · pdf; median \(m\) · Regression
CalculusFirst principles · Basic derivatives · Trig derivatives · More derivatives · Inverse trig · Chain, product, quotient · and 12 moreImplicit · Related rates · Tangent; normal · Max / min; inflexion · L'Hôpital, for \(\frac00\) or \(\frac\infty\infty\) · \(\sin^2x\), \(\cos^2x\) · and 3 more

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

Everything in SL AA, plus…

Complex numbers — polar (Cartesian → polar)
\(z = a + bi = r\,\mathrm{cis}\,\theta = re^{i\theta}\)
Modulus & argument ★ not in the booklet
\(r = |z| = \sqrt{a^2 + b^2},\ \theta = \arg z\)
Multiplication / division in polar form ★ not in the booklet
\(z_1 z_2 = r_1 r_2 \,\mathrm{cis}(\theta_1 + \theta_2),\ \tfrac{z_1}{z_2} = \tfrac{r_1}{r_2}\,\mathrm{cis}(\theta_1 - \theta_2)\)
De Moivre's theorem
\(\bigl(r\,\mathrm{cis}\,\theta\bigr)^n = r^n\,\mathrm{cis}(n\theta)\)
Euler's identity ★ not in the booklet
\(e^{i\pi} + 1 = 0\)

Topic 1 (HL only) — Algebra extension

Sum of an infinite geometric series
\(S_\infty = \dfrac{u_1}{1 - r},\ |r| < 1\)
Proof by induction (skeleton) ★ not in the booklet
\(P(1)\text{ true};\ P(k) \Rightarrow P(k+1);\ \therefore P(n)\text{ true for all }n\in\mathbb{Z}^+\)

Practise Topic 1: free HL AA questions with mark schemes →

Using the booklet: a worked example

Find the exact value of \(\displaystyle\int_0^1 x e^{2x}\,dx\).

  1. From the booklet, \(\int u \dfrac{dv}{dx}\,dx = uv - \int v \dfrac{du}{dx}\,dx\). Take \(u = x\) and \(\dfrac{dv}{dx} = e^{2x}\), so \(\dfrac{du}{dx} = 1\) and \(v = \tfrac12 e^{2x}\).
  2. \(\int x e^{2x}\,dx = \tfrac12 x e^{2x} - \int \tfrac12 e^{2x}\,dx = \tfrac12 x e^{2x} - \tfrac14 e^{2x} + C\).
  3. \(\Bigl[\tfrac14 e^{2x}(2x - 1)\Bigr]_0^1 = \tfrac14 e^{2} - \left(-\tfrac14\right) = \dfrac{e^2 + 1}{4}\).
  4. Check on the GDC: \(\dfrac{e^2+1}{4} = 2.097\ldots\), the same as the numerical integral.

Integration by parts explained · Free HL AA integration questions with mark schemes

Topic 3 (HL only) — Geometry & Vectors

Vector magnitude
\(|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}\)
Dot product
\(\vec{a}\cdot\vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3 = |\vec{a}||\vec{b}|\cos\theta\)
Cross product (magnitude)
\(|\vec{a}\times\vec{b}| = |\vec{a}||\vec{b}|\sin\theta\)
Cross product (component)
\(\vec{a}\times\vec{b} = (a_2 b_3 - a_3 b_2,\, a_3 b_1 - a_1 b_3,\, a_1 b_2 - a_2 b_1)\)
Vector equation of a line
\(\vec{r} = \vec{a} + t\,\vec{d}\)
Cartesian equation of a plane
\(\vec{r}\cdot\vec{n} = \vec{a}\cdot\vec{n}\)
Compound-angle identities
\(\sin(A\pm B) = \sin A\cos B \pm \cos A\sin B\)
Compound-angle (cosine)
\(\cos(A\pm B) = \cos A\cos B \mp \sin A\sin B\)

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

Topic 4 (HL only) — Statistics & Probability extension

Bayes' theorem
\(P(A \mid B) = \dfrac{P(B \mid A) P(A)}{P(B)}\)
Variance (continuous)
\(\mathrm{Var}(X) = E(X^2) - \bigl[E(X)\bigr]^2\)
Linearity of expectation
\(E(aX + b) = aE(X) + b,\ \mathrm{Var}(aX + b) = a^2 \mathrm{Var}(X)\)

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

Topic 5 (HL only) — Calculus extension

Derivative of tan x
\(\dfrac{d}{dx}(\tan x) = \sec^2 x\)
Integration by parts
\(\int u\, dv = uv - \int v\, du\)
Standard integrals (HL)
\(\int \dfrac{1}{\sqrt{a^2-x^2}}\, dx = \arcsin\!\tfrac{x}{a} + C,\ \int \dfrac{1}{a^2 + x^2}\, dx = \tfrac{1}{a}\arctan\!\tfrac{x}{a} + C\)
Maclaurin series
\(f(x) = f(0) + f'(0)x + \dfrac{f''(0)}{2!}x^2 + \dfrac{f'''(0)}{3!}x^3 + \cdots\)
Standard Maclaurin expansions
\(e^x = \sum_{k=0}^{\infty}\tfrac{x^k}{k!},\ \sin x = \sum_{k=0}^{\infty}\tfrac{(-1)^k x^{2k+1}}{(2k+1)!},\ \cos x = \sum_{k=0}^{\infty}\tfrac{(-1)^k x^{2k}}{(2k)!}\)
Volume of revolution (about x-axis)
\(V = \pi \int_a^b y^2\, dx\)
Volume of revolution (about y-axis)
\(V = \pi \int_c^d x^2\, dy\)

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

The SL formulas (also in the HL booklet)

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

Prior learning

Area of a parallelogram
\(A = b \cdot h\)
Area of a triangle
\(A = \tfrac{1}{2}(b \cdot h)\)
Area of a trapezium
\(A = \tfrac{1}{2}(a+b) h\)
Area of a circle
\(A = \pi r^2\)
Volume of a cuboid
\(V = l w h\)
Volume of a cylinder
\(V = \pi r^2 h\)
Distance between two points
\(d = \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}\)
Midpoint of two points
\(\left(\tfrac{x_1+x_2}{2},\ \tfrac{y_1+y_2}{2}\right)\)

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
\(S_n = \tfrac{n}{2}\bigl(2u_1 + (n-1)d\bigr) = \tfrac{n}{2}(u_1 + u_n)\)
The n-th term of a geometric sequence
\(u_n = u_1 \cdot r^{\,n-1}\)
Sum of n terms of a geometric sequence
\(S_n = \frac{u_1(r^n - 1)}{r - 1},\ r \ne 1\)
Infinite geometric series
\(S_\infty = \dfrac{u_1}{1 - r},\ |r| < 1\)
Compound interest
\(FV = PV\left(1 + \frac{r}{100k}\right)^{kn}\)
Exponents & logs (change of base)
\(\log_a x = \dfrac{\log_b x}{\log_b a}\)
Product / quotient / power laws
\(\log_a(xy) = \log_a x + \log_a y,\ \log_a\!\tfrac{x}{y} = \log_a x - \log_a y,\ \log_a(x^n) = n\log_a x\)
Binomial theorem
\((a+b)^n = \sum_{k=0}^{n}\binom{n}{k} a^{n-k} b^{k},\quad \binom{n}{k} = \dfrac{n!}{k!(n-k)!}\)

Topic 2 — Functions

Equation of a straight line
\(y - y_1 = m(x - x_1)\)
Gradient
\(m = \dfrac{y_2 - y_1}{x_2 - x_1}\)
Axis of symmetry of a quadratic
\(x = -\dfrac{b}{2a}\)
Quadratic formula
\(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
Discriminant
\(\Delta = b^2 - 4ac\)

Topic 3 — Geometry & Trigonometry

Arc length (radians)
\(\ell = r\theta\)
Area of a sector (radians)
\(A = \tfrac{1}{2} r^2 \theta\)
Sine rule
\(\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}\)
Cosine rule
\(c^2 = a^2 + b^2 - 2ab\cos C\)
Area of a triangle
\(A = \tfrac{1}{2}ab\sin C\)
Pythagorean identity
\(\sin^2\theta + \cos^2\theta = 1\)
Double-angle: sine
\(\sin 2\theta = 2\sin\theta\cos\theta\)
Double-angle: cosine
\(\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta\)
Tangent from sine & cosine
\(\tan\theta = \dfrac{\sin\theta}{\cos\theta}\)

Topic 4 — Statistics & Probability

Interquartile range
\(\mathrm{IQR} = Q_3 - Q_1\)
Mean of a data set
\(\bar{x} = \dfrac{\sum f_i x_i}{n},\ n = \sum f_i\)
Probability of an event
\(P(A) = \dfrac{n(A)}{n(U)}\)
Complementary events
\(P(A') = 1 - P(A)\)
Combined events
\(P(A \cup B) = P(A) + P(B) - P(A \cap B)\)
Conditional probability
\(P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}\)
Independent events
\(P(A \cap B) = P(A)P(B)\)
Expected value (discrete)
\(E(X) = \sum x P(X=x)\)
Binomial distribution
\(X \sim B(n,p) \implies P(X=r) = \binom{n}{r} p^r (1-p)^{n-r}\)
Mean & variance of a binomial
\(E(X) = np,\quad \mathrm{Var}(X) = np(1-p)\)
Standardising a normal variable
\(Z = \dfrac{X - \mu}{\sigma}\)

Topic 5 — Calculus

Derivative of xⁿ
\(\dfrac{d}{dx}\bigl(x^n\bigr) = n x^{\,n-1}\)
Chain rule
\(\dfrac{dy}{dx} = \dfrac{dy}{du} \cdot \dfrac{du}{dx}\)
Product rule
\((uv)' = u'v + uv'\)
Quotient rule
\(\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}\)
Derivatives of standard functions
\(\dfrac{d}{dx}(\sin x) = \cos x,\ \dfrac{d}{dx}(\cos x) = -\sin x\)
Exponential & log derivatives
\(\dfrac{d}{dx}(e^x) = e^x,\ \dfrac{d}{dx}(\ln x) = \dfrac{1}{x}\)
Integral of xⁿ
\(\int x^n\, dx = \dfrac{x^{n+1}}{n+1} + C,\ n \ne -1\)
Standard integrals
\(\int \sin x\, dx = -\cos x + C,\ \int \cos x\, dx = \sin x + C,\ \int e^x\, dx = e^x + C,\ \int \dfrac{1}{x}\, dx = \ln|x| + C\)
Area between a curve and the x-axis
\(A = \int_a^b y\, dx\)

Try it now

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5 quick questions from across HL AA. No sign-in needed.

Find the sum to infinity of the geometric series with first term \(18\) and common ratio \(\frac{2}{3}\).
Choose your answer · keys 1–4

Not in the AA HL booklet: 42 results to learn

The IB Maths AA 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

Indices
\(a^ma^n=a^{m+n},\quad (a^m)^n=a^{mn},\quad a^{-n}=\tfrac1{a^n},\quad a^{\frac mn}=\sqrt[n]{a^m}\)

Number & algebra

\((a+bx)^n=a^n\left(1+\tfrac bax\right)^n\), valid for \(|x|<\left|\tfrac ab\right|\)
Partial fractions
\(\frac{px+q}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b}\)
Modulus, argument, conjugate
\(r=|z|=\sqrt{a^2+b^2},\quad \tan\theta=\tfrac ba,\quad z^*=a-bi,\quad zz^*=|z|^2\)
Products
\(|z_1z_2|=|z_1||z_2|,\quad \arg(z_1z_2)=\arg z_1+\arg z_2,\quad \arg\tfrac{z_1}{z_2}=\arg z_1-\arg z_2\)
\(n\)th roots of \(re^{i\theta}\)
\(r^{1/n}e^{i(\theta+2\pi k)/n},\quad k=0,1,\dots,n-1\)
For \(|z|=1\)
\(z^n+z^{-n}=2\cos n\theta,\quad z^n-z^{-n}=2i\sin n\theta\)
Roots of unity sum to 0; non-real roots of real polynomials come in conjugate pairs
Induction: prove \(P(1)\); assume \(P(k)\); prove \(P(k+1)\); conclude for all \(n\in\mathbb Z^+\)
Contradiction: assume the negation, reach a contradiction. Counterexample disproves

Functions

Perpendicular
\(m_1m_2=-1\)
\(\Delta>0\): two real roots; \(\Delta=0\): repeated root; \(\Delta<0\): complex conjugate roots
Factor & remainder theorems
\(p(a)=0\iff(x-a)\text{ is a factor};\quad \text{remainder of }p(x)\div(x-a)\text{ is }p(a)\)
Odd; even
\(f(-x)=-f(x);\quad f(-x)=f(x)\)
Composite; inverse
\((f\circ g)(x)=f(g(x));\quad f\big(f^{-1}(x)\big)=x\)
Transformations: \(f(x)+b\) up \(b\); \(f(x-a)\) right \(a\); \(pf(x)\) vertical ×\(p\); \(f(qx)\) horizontal ×\(\tfrac1q\); \(-f(x)\), \(f(-x)\) reflect in \(x\)-, \(y\)-axis
Graphs: \(|f(x)|\) reflects negative parts up; \(f(|x|)\) mirrors \(x\ge0\); \(\frac1{f(x)}\) has asymptotes at zeros of \(f\)
\(y=\frac{ax+b}{cx+d}\) asymptotes
\(x=-\frac dc,\quad y=\frac ac\)

Geometry & trigonometry

Exact values \(0,\frac\pi6,\frac\pi4,\frac\pi3,\frac\pi2\)
\(\sin:0,\tfrac12,\tfrac{\sqrt2}2,\tfrac{\sqrt3}2,1;\quad \cos\text{: reverse};\quad \tan:0,\tfrac1{\sqrt3},1,\sqrt3,\text{undef.}\)
Ranges
\(\arcsin x\in[-\tfrac\pi2,\tfrac\pi2],\quad \arccos x\in[0,\pi],\quad \arctan x\in(-\tfrac\pi2,\tfrac\pi2)\)
\(a\sin(b(x-c))+d\), \(b>0\)
\(\text{amplitude }|a|,\quad \text{period }\tfrac{2\pi}b\)
Perpendicular; unit vector; \(\overrightarrow{AB}\)
\(\boldsymbol v\cdot\boldsymbol w=0;\quad \hat{\boldsymbol v}=\frac{\boldsymbol v}{|\boldsymbol v|};\quad \boldsymbol b-\boldsymbol a\)
Area of triangle
\(\tfrac12|\boldsymbol v\times\boldsymbol w|\)
Angle line–plane; plane–plane
\(\sin\theta=\frac{|\boldsymbol b\cdot\boldsymbol n|}{|\boldsymbol b||\boldsymbol n|};\quad \cos\theta=\frac{|\boldsymbol n_1\cdot\boldsymbol n_2|}{|\boldsymbol n_1||\boldsymbol n_2|}\)
Normal to plane through \(\boldsymbol a\) with directions \(\boldsymbol b,\boldsymbol c\)
\(\boldsymbol n=\boldsymbol b\times\boldsymbol c\)
Line of intersection of two planes: direction \(\boldsymbol n_1\times\boldsymbol n_2\)

Statistics & probability

Outliers
\(<Q_1-1.5\,IQR\quad \text{or}\quad >Q_3+1.5\,IQR\)
Variance
\(\sigma^2=\frac{\sum f(x-\mu)^2}{n}=\frac{\sum fx^2}{n}-\mu^2\)
Binomial pmf
\(P(X=r)=\tbinom nrp^r(1-p)^{n-r}\)
pdf; median \(m\)
\(\int f(x)\,dx=1;\quad \int_{-\infty}^mf(x)\,dx=\tfrac12;\quad \text{mode: max of }f\)
Regression: \(y\) on \(x\) to predict \(y\), \(x\) on \(y\) for \(x\); interpolate only

Calculus

Implicit
\(\tfrac{d}{dx}\big(y^n\big)=ny^{n-1}\tfrac{dy}{dx},\quad \tfrac{d}{dx}(xy)=x\tfrac{dy}{dx}+y\)
Related rates
\(\frac{dV}{dt}=\frac{dV}{dr}\cdot\frac{dr}{dt}\)
Tangent; normal
\(y-f(a)=f'(a)(x-a);\quad m_\text{n}=-\tfrac1{f'(a)}\)
Max / min; inflexion
\(f'=0\text{ and }f''<0\ /\ f''>0;\quad f''=0\text{ with sign change}\)
L'Hôpital, for \(\frac00\) or \(\frac\infty\infty\)
\(\lim\frac{f(x)}{g(x)}=\lim\frac{f'(x)}{g'(x)}\)
Learn
\(\int\sec^2x\,dx=\tan x+C,\quad \int\tfrac{f'(x)}{f(x)}\,dx=\ln|f(x)|+C,\quad \int f(ax+b)\,dx=\tfrac1aF(ax+b)+C\)
\(\sin^2x\), \(\cos^2x\)
\(\sin^2x=\tfrac12(1-\cos2x),\quad \cos^2x=\tfrac12(1+\cos2x)\)
Substitution
\(\int f(g(x))g'(x)\,dx=\int f(u)\,du;\quad \text{change the limits too}\)
Separable
\(\frac{dy}{dx}=f(x)g(y)\Rightarrow\int\frac{dy}{g(y)}=\int f(x)\,dx\)
Homogeneous \(\frac{dy}{dx}=f\big(\frac yx\big)\)
\(y=vx,\quad \frac{dy}{dx}=v+x\frac{dv}{dx}\)
Then
\(y\,e^{\int P\,dx}=\int Q\,e^{\int P\,dx}\,dx\)

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

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

Calculator tip: Paper 1 has no calculator. In Papers 2 and 3, check a definite integral or the roots of an equation on the GDC, and when \(\mu\) or \(\sigma\) is unknown use inverse normal with \(\mu = 0\) and \(\sigma = 1\) to get \(z\), then solve \(z = \dfrac{x-\mu}{\sigma}\).

IB Maths AA 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 AA 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

Derivative of sin x Standard deviation Volume of a cone Volume of a sphere

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 AA HL PDF has both on one page.

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

Yes. A clean copy of the formula booklet is provided in every paper: calculator-free Paper 1, and Papers 2 and 3. 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. AA 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 AA 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 AA 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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