Percentage change
Increase or decrease relative to an original value.
Change (%) = ((New − Original) / Original) × 100 Cambridge International A Level 9709
🧮 Cambridge International A Level Maths Formula Sheet 2026
Pure maths, mechanics and statistics formulas aligned to the Cambridge 9709 syllabus — summarised for Paper 1–6 preparation.
Pure Mathematics Mechanics Statistics
Our formula sheets are free to download — save this one as PDF for offline revision.
Aligned with the latest 2026 syllabus and board specifications. This sheet is prepared to match your exam board’s official specifications for the 2026 exam series.
Everything You Need for Cambridge A Level Mathematics
Whether you are sitting the AS or full A Level, this formula sheet organises differentiation, integration, series, vectors, kinematics and probability formulas with short reminders to help you avoid common mistakes.
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Pure maths identities with exam-ready notes
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Mechanics equations with vector form reminders
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Statistics formulas for discrete and continuous models
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Tips to pair formulas with Cambridge command words
Number, ratio & proportion
Percentages, growth, standard measures, and proportional reasoning used across GCSE papers.
Simple interest
P principal, R annual % rate, T years.
Interest = (P × R × T) / 100 Compound interest & growth
P principal, r decimal rate per period, n periods per year, t years.
A = P(1 + r/n)^(nt) Ratio & proportion
Equivalent ratios and cross-multiplication.
a : b = c : d ⟺ a/b = c/d ⟹ ad = bc Speed, distance, time
Consistent units (e.g. m/s, km/h).
speed = distance / time distance = speed × time Density
Mass m, volume V.
density = mass / volume Pressure (force & area)
F force, A area.
pressure = force / area Direct proportion
y proportional to x.
y = kx (k constant) Inverse proportion
y inversely proportional to x.
y = k/x or xy = k Topic Focus
Exam tips
- Convert mixed units before substituting (e.g. km → m, hours → minutes).
- For reverse percentage problems, identify whether the ‘new’ value is after an increase or decrease.
- Show working: method marks often depend on a correct ratio or percentage set-up.
Algebra, graphs & sequences
Linear and quadratic relationships, coordinate facts, and common sequence formulae.
Straight-line graph
m gradient, c y-intercept.
y = mx + c Gradient from two points
Points (x₁, y₁) and (x₂, y₂).
m = (y₂ − y₁) / (x₂ − x₁) Perpendicular lines
Gradients m₁ and m₂ (neither vertical/horizontal mismatch).
m₁ × m₂ = −1 Midpoint of a line segment
Between (x₁, y₁) and (x₂, y₂).
((x₁ + x₂)/2 , (y₁ + y₂)/2) Distance between two points
In the coordinate plane.
d = √[(x₂ − x₁)² + (y₂ − y₁)²] Quadratic formula
Roots of ax² + bx + c = 0, a ≠ 0.
x = [−b ± √(b² − 4ac)] / (2a) Difference of two squares
Factorising.
a² − b² = (a − b)(a + b) Arithmetic sequence — nth term
First term a, common difference d.
uₙ = a + (n − 1)d Arithmetic series — sum of n terms
Last term l optional.
Sₙ = n/2 [2a + (n − 1)d] Sₙ = n/2 (a + l) Geometric sequence — nth term
First term a, common ratio r.
uₙ = arⁿ⁻¹ Geometric series — sum of n terms
r ≠ 1.
Sₙ = a(1 − rⁿ) / (1 − r) Circle in the plane
Centre (a, b), radius r.
(x − a)² + (y − b)² = r² Topic Focus
Graph & algebra
- Parallel lines share the same gradient; perpendicular gradients multiply to −1.
- For quadratics, link discriminant b² − 4ac to number of real roots.
- Check sequence type: constant first difference → arithmetic; constant ratio → geometric.
Geometry — perimeter, area & circles
Plane shapes, compound figures, and circle measures.
Rectangle
Length l, width w.
Area = lw Perimeter = 2l + 2w Triangle
Base b, perpendicular height h.
Area = ½ × b × h Parallelogram
Base b, perpendicular height h.
Area = b × h Trapezium
Parallel sides a and b, perpendicular height h.
Area = ½ (a + b) × h Circle
Radius r, diameter d = 2r.
Area = πr² Circumference = 2πr = πd Arc length
Angle θ° at centre.
Arc = (θ/360) × 2πr Sector area
Angle θ° at centre.
Sector area = (θ/360) × πr² Topic Focus
Mensuration
- Height must be perpendicular to the chosen base.
- For composite shapes, split into standard shapes and add or subtract areas.
- Keep angles in degrees consistent with arc/sector formulae above.
Geometry — volume & surface area
Prisms, cylinders, cones, spheres, and pyramids.
Prism
Cross-sectional area A, length l.
Volume = A × l Cylinder
Radius r, height h.
Volume
V = πr²h Curved surface area
2πrh Cone
Radius r, height h, slant height l.
Volume
V = ⅓ πr²h Curved surface area
πrl Link
l² = r² + h² (Pythagoras) Sphere
Radius r.
Volume
V = (4/3)πr³ Surface area
A = 4πr² Pyramid
Base area A, perpendicular height h.
V = ⅓ × A × h Pythagoras’ theorem
Right-angled triangle, hypotenuse c.
a² + b² = c² Topic Focus
3D problems
- Volume of prism = uniform cross-section × length (or height along the prism).
- Cone and pyramid volumes both use the factor one-third.
- Use Pythagoras in 3D to find slant heights and diagonals when needed.
Trigonometry
Right-angled triangles, sine/cosine rules, and triangle area.
SOHCAHTOA (right-angled triangles)
Angle θ opposite, adjacent, hypotenuse.
sin θ = opposite / hypotenuse cos θ = adjacent / hypotenuse tan θ = opposite / adjacent Sine rule
Any triangle with sides a, b, c opposite angles A, B, C.
a / sin A = b / sin B = c / sin C Cosine rule
Finding a side or an angle.
a² = b² + c² − 2bc cos A cos A = (b² + c² − a²) / (2bc) Area of a triangle (two sides and included angle)
Sides b, c and included angle A.
Area = ½ bc sin A Topic Focus
Choosing a method
- Right-angled only: SOHCAHTOA or Pythagoras.
- Non-right: sine rule when you have a matching side/angle pair; cosine rule for SAS or SSS.
- Always check your calculator is in degree mode for GCSE unless a question specifies radians.
Statistics & probability
Summaries of data, probability rules, and expectation.
Mean
Arithmetic average.
mean = (sum of values) / (number of values) Estimated mean (grouped data)
Midpoints mᵢ, frequencies fᵢ.
≈ Σ(mᵢ × fᵢ) / Σfᵢ Range
Spread of data.
range = largest value − smallest value Median & mode
Median: middle value when ordered; mode: most frequent.
If there is an even count of values, median is the mean of the two middle values.
Probability of an event
Equally likely outcomes.
P(A) = (number of outcomes for A) / (total possible outcomes) Complement
Probability of not A.
P(not A) = 1 − P(A) Independent events
A and B independent.
P(A and B) = P(A) × P(B) Expected frequency
Trials n, probability p of success.
expected frequency ≈ n × p Topic Focus
Data & chance
- Mean uses all values and is pulled by outliers; median is often better for skewed data.
- Probabilities lie between 0 and 1; the sum of probabilities of all mutually exclusive outcomes is 1.
- For tree diagrams, multiply along branches for combined events.
Kinematics (constant acceleration)
suvat relationships used in GCSE Mathematics and linked contexts; u initial velocity, v final, a acceleration, s displacement, t time.
Definitions
Straight-line motion with constant acceleration.
v = u + at s = ½(u + v)t s = ut + ½at² v² = u² + 2as Topic Focus
Using suvat
- Use consistent SI units: m, s, m/s, m/s².
- Define a positive direction first; negative acceleration means slowing in the positive direction.
- Identify which quantity is missing and pick the equation that contains only knowns and that unknown.
Differentiation
Gradients, tangents, and optimisation.
First principles
f′(x) = lim_{h→0} (f(x+h) − f(x)) / h Common derivatives
d/dx (x^n) = n x^{n−1} d/dx (e^x) = e^x d/dx (ln x) = 1/x d/dx (sin x) = cos x d/dx (cos x) = −sin x Chain rule
dy/dx = dy/du × du/dx Product rule
d/dx (u v) = u′ v + u v′ Quotient rule
d/dx (u/v) = (u′ v − u v′) / v² Stationary points
Solve f′(x) = 0; classify with f″(x) or sign change Topic Focus
Modelling
- Interpret derivative as rate of change with correct units.
- Second derivative: concavity and acceleration in kinematics links.
Integration
Antiderivatives, definite integrals, and area.
Fundamental idea
∫ f(x) dx = F(x) + C where F′ = f Definite integral
∫_a^b f(x) dx = F(b) − F(a) Area under curve
Area = ∫_a^b y dx (y ≥ 0) Area between curves
∫_a^b (f − g) dx where f ≥ g Integration by parts
∫ u dv = u v − ∫ v du Substitution
∫ f(u(x)) u′(x) dx = ∫ f(u) du Topic Focus
Interpretation
- Negative definite integral means net signed area below axis.
- For rotation volumes at FM: disc/washer methods as per specification.
Trigonometry & identities
Pythagorean type identities and equations.
Pythagorean
sin² θ + cos² θ = 1 1 + tan² θ = sec² θ Angle addition
sin(A ± B) = sin A cos B ± cos A sin B cos(A ± B) = cos A cos B ∓ sin A sin B Double angle
sin 2θ = 2 sin θ cos θ cos 2θ = cos² θ − sin² θ R–a method
a sin x ± b cos x.
R sin(x ± α) or R cos(x ± α) Topic Focus
Solving
- Use identities to reduce to single trig type; check extraneous roots in given interval.
- Small-angle approximations: sin θ ≈ θ, cos θ ≈ 1 − θ²/2 (radians).
Vectors (2D & 3D)
Scalar product, equations of lines.
Magnitude
|a| = √(a₁² + a₂² + a₃²) Scalar (dot) product
a · b = |a| |b| cos θ = a₁b₁ + a₂b₂ + a₃b₃ Angle between vectors
cos θ = (a · b) / (|a| |b|) Line (vector form)
r = a + λ d Shortest distance (point to line)
Use perpendicular vector.
Projection and cross product methods per specification Topic Focus
Geometry
- Parallel vectors: d₁ = k d₂; perpendicular: a · b = 0.
- Plane equation: r · n = p · n (FM).
Probability & distributions (A level)
Discrete and continuous models.
Conditional probability
P(A|B) = P(A ∩ B) / P(B) Binomial
n trials, p success probability.
P(X = r) = C(n,r) p^r (1−p)^{n−r} Normal standardising
Z = (X − μ) / σ Approximation
np, n(1−p) large.
Normal approx to binomial with continuity correction when allowed.
Topic Focus
Hypothesis tests
- State H₀/H₁; compare p-value or critical region to significance level.
- Know assumptions for each model (independence, constant p, etc.).
Numerical methods
Root finding and integration estimates.
Newton–Raphson
x_{n+1} = x_n − f(x_n) / f′(x_n) Trapezium rule
∫_a^b y dx ≈ h/2 (y₀ + 2y₁ + … + 2y_{n−1} + y_n) Topic Focus
Iteration
- Check convergence by sign change of f or small step size.
- Trapezium rule biased; Simpson’s rule (FM) improves accuracy.
How to Use This Formula Sheet
Boost your Cambridge exam confidence with these proven study strategies from our tutoring experts.
📘
Quote Formula, Then Apply
Cambridge examiners award method marks when you state the relevant formula before substitution — especially in integration and statistics questions.
🧭
Sketch for Sign Checks
Draw quick graphs or motion diagrams to confirm limits, signs and directions before plugging values into calculus or mechanics formulas.
🧠
Link Papers Together
Pure maths often underpins mechanics or statistics parts. Note which pure topics feed into applied questions to save revision time.
⏳
Rehearse Calculator Workflow
Practise entering regression, normal distribution and vector calculations on your calculator so you can reproduce them quickly in the exam.
Formula sheet FAQ
Quick answers about this free PDF, how to use it for exam revision, and how it relates to your official syllabus.
Is the Cambridge International A Level Maths Formula Sheet 2026 free to download as a PDF?
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What Mathematics topics and equations does this formula sheet cover?
This page groups key Mathematics formulas in one place for revision. Complete Cambridge International A Level Mathematics (9709) formula sheet for 2026 examinations, covering pure maths, calculus, trigonometry, mechanics and statistics with examiner tips. Always cross-check with your official syllabus and past papers for your exam session.
Can I use this instead of the official exam formula booklet in the exam?
No. In the exam you must follow only what your exam board allows in the hall—usually the official formula booklet or data sheet where provided. This page is a revision and teaching aid, not a replacement for board-issued materials.
Who is this formula sheet for (Upper Secondary)?
It is written for students preparing for assessments at Upper Secondary in Mathematics, including classroom revision, homework support, and independent study. Teachers and tutors can also share it as a quick reference.
How should I revise with this formula sheet?
Work through past paper questions, quote the correct formula before substituting values, and check units and notation every time. Pair this sheet with timed practice and mark schemes so you see how examiners expect working to be set out.
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References the Cambridge International AS & A Level Mathematics (9709) syllabus including Pure Mathematics 1–3, Mechanics, and Probability & Statistics.
State assumptions (e.g., light string, particle model, normal approximation conditions) whenever you apply these formulas in long-form answers.