A-level Further Maths formula booklet

Every board gives you a formula booklet in every Further Maths paper. It holds the harder results (series, Maclaurin expansions, hyperbolic identities, conics, distributions, centres of mass) and the statistical tables. It leaves out the basics, and those you must learn: the quadratic formula, laws of logs, double angles, complex numbers in exponential form, the 2 by 2 inverse, sums and products of roots, the hyperbolic definitions, volumes of revolution and the vector line and plane.

What each board's booklet holds

Pearson Edexcel 9FM0
Mathematical formulae and statistical tables
One booklet for AS and A level Maths and Further Maths, given in every paper.
AS and A level Maths
Mensuration, arithmetic, geometric and binomial series, logarithms, trigonometric identities, small angles, differentiation from first principles, numerical methods, probability, standard deviation, distributions and kinematics.
Further Pure
Summations of r² and r³, matrix transformations, area of a sector in polar form, complex numbers, Maclaurin and Taylor series, vectors (including the vector product and planes), hyperbolic identities and inverses, conics, harder derivatives and integrals, arc length and surface area of revolution.
Further Statistics
Discrete and continuous distributions with their means and variances, correlation and regression, expectation algebra, sampling distributions and non-parametric tests.
Further Mechanics
Centres of mass of standard shapes and motion in a circle.
Statistical tables
Binomial and Poisson cumulative probabilities, percentage points of the normal, χ², t and F distributions, critical values for correlation coefficients and random numbers.
The board's booklet: on the Pearson Edexcel website
AQA 7367
Formulae and statistical tables for A-level Mathematics and A-level Further Mathematics
Required in every AS and A-level Further Maths exam; A-level Maths students may use it too.
Pure
Binomial, arithmetic and geometric series, trigonometric identities, differentiation and integration tables, numerical methods, complex numbers, matrix transformations, summations, Maclaurin series, vectors, area of a sector, hyperbolic identities, conics, numerical solution of differential equations, arc length and surface area of revolution.
Mechanics
Constant acceleration and centres of mass.
Probability and statistics
Probability, standard deviation, discrete distributions, sampling distributions and distribution-free tests.
Statistical tables
Percentage points of the t and χ² distributions, and critical values of the product moment correlation coefficient. There is no normal table: use your calculator.
The board's booklet (PDF)
OCR A H245
A Level Further Mathematics A Formulae Booklet
Given in each Further Maths A paper: Pure Core, Statistics, Mechanics, Discrete and Additional Pure.
Pure
Series and Maclaurin series, matrix transformations, differentiation and integration, polar area, numerical methods, complex numbers, trigonometric and hyperbolic identities and simple harmonic motion.
Statistics
Probability, discrete and continuous distributions, sampling, expectation algebra, non-parametric tests, correlation and regression.
Mechanics
Kinematics, Newton's experimental law, motion in a circle and centres of mass.
Discrete and Additional Pure
Network and sorting algorithms, the simplex method, inclusion-exclusion, the vector product and surfaces.
Statistical tables
Percentage points of the normal and χ² distributions, critical values for the product moment and Spearman's rank coefficients, and Wilcoxon signed-rank and rank-sum tables.
The board's booklet: on the OCR website

Further Maths formulas to learn: not in the booklet

The same list for Edexcel, AQA and OCR, except where a tag says otherwise. None of it is printed in the booklet.

Algebra and series

Quadratic formula: $\displaystyle ax^2 + bx + c = 0 \;\Rightarrow\; x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Laws of indices: $\displaystyle a^x a^y = a^{x+y}$ $\displaystyle a^x \div a^y = a^{x-y}$ $\displaystyle (a^x)^y = a^{xy}$
Laws of logarithms: $\displaystyle \log_a x + \log_a y = \log_a xy$ $\displaystyle \log_a x - \log_a y = \log_a \tfrac{x}{y}$ $\displaystyle k \log_a x = \log_a x^k$ $x = a^n \iff n = \log_a x$ , for $a > 0$ and $x > 0$ .
General terms: $\displaystyle u_n = a + (n-1)d$ $\displaystyle u_n = ar^{n-1}$ Arithmetic, then geometric.
Sum of the first n integers: $\displaystyle \sum_{r=1}^{n} r = \tfrac{1}{2}n(n+1)$ The sums of $r^2$ and $r^3$ are in the booklet.
Roots of a quadratic: $\displaystyle \alpha + \beta = -\frac{b}{a}$ $\displaystyle \alpha\beta = \frac{c}{a}$
Roots of a cubic: $\displaystyle \sum \alpha = -\frac{b}{a}$ $\displaystyle \sum \alpha\beta = \frac{c}{a}$ $\displaystyle \alpha\beta\gamma = -\frac{d}{a}$ For $ax^3 + bx^2 + cx + d = 0$ .

Coordinates, trigonometry and mensuration

Straight lines: $\displaystyle y - y_1 = m(x - x_1)$ Perpendicular when $m_1 m_2 = -1$ .
Sine and cosine rules, area: $\displaystyle \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ $\displaystyle a^2 = b^2 + c^2 - 2bc\cos A$ $\displaystyle \text{Area} = \tfrac{1}{2}ab\sin C$
Pythagorean identities: $\displaystyle \cos^2 A + \sin^2 A = 1$ $\displaystyle \sec^2 A = 1 + \tan^2 A$ $\displaystyle \operatorname{cosec}^2 A = 1 + \cot^2 A$
Double angles: $\displaystyle \sin 2A = 2\sin A\cos A$ $\displaystyle \cos 2A = \cos^2 A - \sin^2 A$ $\displaystyle \tan 2A = \frac{2\tan A}{1 - \tan^2 A}$
Circles, arcs and sectors: $\displaystyle C = 2\pi r$ $\displaystyle A = \pi r^2$ $\displaystyle s = r\theta$ $\displaystyle A = \tfrac{1}{2}r^2\theta$ $\theta$ in radians.
Trapezium and prism: $\displaystyle \text{Area} = \tfrac{1}{2}(a + b)h$ $\displaystyle \text{Volume} = \text{cross-section area} \times \text{length}$

Complex numbers and matrices

Multiplying and dividing: $\displaystyle z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)}$ $\displaystyle \frac{z_1}{z_2} = \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)}$
Exponential form: $\displaystyle e^{i\theta} = \cos\theta + i\sin\theta$
Loci: $|z - a| = r$ is a circle, centre $a$ , radius $r$ . $\arg(z - a) = \theta$ is a half-line from $a$ at angle $\theta$ to the positive real direction.
2 by 2 determinant and inverse: $\displaystyle \Delta = ad - bc$ $\displaystyle \begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{\Delta}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$
Products of matrices: $\displaystyle (\mathbf{AB})^{-1} = \mathbf{B}^{-1}\mathbf{A}^{-1}$ $\mathbf{AB}$ means the transformation $\mathbf{B}$ first, then $\mathbf{A}$ .

Hyperbolic functions and calculus

Hyperbolic functions: $\displaystyle \cosh x = \tfrac{1}{2}(e^x + e^{-x})$ $\displaystyle \sinh x = \tfrac{1}{2}(e^x - e^{-x})$ $\displaystyle \tanh x = \frac{\sinh x}{\cosh x}$
Standard derivatives: $\displaystyle x^n \to nx^{n-1}$ $\displaystyle \sin kx \to k\cos kx$ $\displaystyle \cos kx \to -k\sin kx$ $\displaystyle e^{kx} \to ke^{kx}$ $\displaystyle \ln x \to \tfrac{1}{x}$ With the sum, product and chain rules.
Hyperbolic derivatives and integralsEdexcel and OCR: $\displaystyle \frac{d}{dx}\sinh kx = k\cosh kx$ $\displaystyle \frac{d}{dx}\cosh kx = k\sinh kx$ $\displaystyle \int \cosh kx\,dx = \tfrac{1}{k}\sinh kx + c$
Standard integrals: $\displaystyle \int x^n\,dx = \frac{x^{n+1}}{n+1} + c \;(n \ne -1)$ $\displaystyle \int \tfrac{1}{x}\,dx = \ln|x| + c$ $\displaystyle \int e^{kx}\,dx = \tfrac{1}{k}e^{kx} + c$ And $\int \cos kx\,dx = \tfrac{1}{k}\sin kx + c$ , $\int \sin kx\,dx = -\tfrac{1}{k}\cos kx + c$ .
Area and volumes of revolution: $\displaystyle \text{Area} = \int_a^b y\,dx$ $\displaystyle V_x = \pi\int_a^b y^2\,dx$ $\displaystyle V_y = \pi\int_c^d x^2\,dy$
Simple harmonic motion: $\displaystyle \ddot{x} = -\omega^2 x$

Vectors

Magnitude: $\displaystyle |x\mathbf{i} + y\mathbf{j} + z\mathbf{k}| = \sqrt{x^2 + y^2 + z^2}$
Scalar product: $\displaystyle \mathbf{a}\cdot\mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 = |\mathbf{a}||\mathbf{b}|\cos\theta$
Line and plane: $\displaystyle \mathbf{r} = \mathbf{a} + t\mathbf{b}$ $\displaystyle (\mathbf{r} - \mathbf{a})\cdot\mathbf{n} = 0$

Mechanics and statistics

Forces: $\displaystyle W = mg$ $\displaystyle F \le \mu R$ $\displaystyle F = ma$
Variable acceleration: $\displaystyle v = \frac{dr}{dt}$ $\displaystyle a = \frac{dv}{dt} = \frac{d^2r}{dt^2}$ $\displaystyle r = \int v\,dt$ $\displaystyle v = \int a\,dt$
Momentum and energyEdexcel only: $\displaystyle \text{momentum} = mv$ $\displaystyle \text{impulse} = mv - mu$ $\displaystyle \text{KE} = \tfrac{1}{2}mv^2$ $\displaystyle \text{PE} = mgh$
Elastic stringsEdexcel only: $\displaystyle T = \frac{\lambda x}{l}$ $\displaystyle \text{EPE} = \frac{\lambda x^2}{2l}$
Solving simple harmonic motionEdexcel only: $\displaystyle x = a\cos\omega t \text{ or } a\sin\omega t$ $\displaystyle v^2 = \omega^2(a^2 - x^2)$ $\displaystyle T = \frac{2\pi}{\omega}$
Mean and the standard normal: $\displaystyle \bar{x} = \frac{\sum x}{n} = \frac{\sum fx}{\sum f}$ $\displaystyle Z = \frac{X - \mu}{\sigma}$

alevelgenius.co.uk/alevel-further-maths/a-level-further-maths-formula-booklet · our own wording, checked October 2026

Test your recall

Twelve formulas from the list, three choices each. Each wrong choice says why.

Question 1 of 12 · Algebra and series

Which is the sum of the first n positive integers?

More free Further Maths help

Questions students ask

Do you get a formula booklet in A-level Further Maths?

Yes. Every board gives one in every paper: Edexcel's Mathematical formulae and statistical tables, AQA's Formulae and statistical tables, and OCR's Further Mathematics A Formulae Booklet.

Is the Further Maths booklet the same as the Maths one?

For Edexcel, yes: one booklet covers AS and A level Maths and Further Maths. AQA's Further Maths booklet also covers A-level Maths, and AQA has a smaller one for Maths only. OCR's Further Maths A booklet is its own.

Which formulas are not in the booklet?

The A-level Maths basics (the quadratic formula, laws of indices and logs, double angles), complex numbers in exponential form, the 2 by 2 inverse, the sum of r, the sums and products of roots, the hyperbolic definitions, volumes of revolution, the scalar product, the vector line and plane, and the mechanics basics. Edexcel adds momentum, energy, elastic strings and the SHM solutions.

Are the sums of r² and r³ in the booklet?

Yes, in all three boards' booklets. The sum of r itself, ½n(n + 1), is not: you learn it.

Can I print the list of formulas to learn?

Yes. Press Print or save as PDF on the sheet: it prints on its own, without the quiz.