Percentage change
Increase or decrease relative to an original value.
Change (%) = ((New − Original) / Original) × 100 Cambridge International A Level 9231
🧮 Cambridge International A Level Further Mathematics Formula Sheet 2026
Further pure mathematics plus common mechanics and statistics relationships for CAIE 9231 — organised for Paper 3/4 style preparation alongside your 9709 foundation.
9231 Further Pure Mechanics & Stats
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.
Further Mathematics for Cambridge International
Cambridge 9231 builds on the techniques in 9709. This sheet layers the further pure content you will need alongside a full single A-level toolkit so you can see how topics connect across papers.
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Proof, complex numbers & matrices
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Polar coordinates & hyperbolic functions
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Differential equations & Maclaurin series
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Further mechanics & statistics relationships
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.
Proof & structure
Proof by induction, contradiction, and counterexamples.
Induction template
Base case; assume P(k); show P(k) ⇒ P(k+1); conclude ∀n ≥ n₀.
Summation
Σ_{r=1}^{n} r = n(n+1)/2; Σ r² = n(n+1)(2n+1)/6; Σ r³ = [n(n+1)/2]² Method of differences
Telescoping sums for partial fractions sequences.
Counterexample
Single case disproving a universal claim.
Topic Focus
Induction
- Match the inductive step to the exact statement P(n).
- Series: often relate S_{k+1} to S_k + T_{k+1}.
Complex numbers
Cartesian, polar, De Moivre, and roots of unity.
Modulus–argument
z = x + iy = r(cos θ + i sin θ) = r e^{iθ} Modulus & conjugate
|z| = √(x² + y²) z* = x − iy z z* = |z|² De Moivre
(cos θ + i sin θ)^n = cos nθ + i sin nθ nth roots of unity
e^{2π i k/n}, k = 0,…,n−1 Roots of polynomials
Complex roots of real coefficients occur in conjugate pairs.
Topic Focus
Loci
- |z − a| = r: circle centre a, radius r.
- arg(z − a) = α: half-line from a.
Matrices (linear algebra)
Eigenvalues, eigenvectors, and diagonalisation (FM core pure).
Characteristic equation
det(A − λI) = 0 Eigenvector
Non-zero v with Av = λv.
2×2 determinant & inverse
det = ad − bc; A⁻¹ = (1/det) [[d, −b], [−c, a]] Transformation
Columns of matrix are images of basis vectors.
Composition
Apply right matrix first: BA means A then B in column-vector convention. Topic Focus
Checks
- Trace = sum of eigenvalues (2×2 with distinct eigenvalues).
- det A = product of eigenvalues.
Further calculus
Improper integrals, volumes of revolution, and hyperbolic functions.
Volume of revolution
π ∫ y² dx π ∫ x² dy Hyperbolic definitions
cosh x = (e^x + e^{−x})/2 sinh x = (e^x − e^{−x})/2 cosh² x − sinh² x = 1 Derivatives
d/dx (sinh x) = cosh x d/dx (cosh x) = sinh x Inverse hyperbolic
arsinh x = ln(x + √(x² + 1)) arcosh x = ln(x + √(x² − 1)), x ≥ 1 Improper integrals
Limits at infinity or at discontinuities; compare tests for convergence.
Topic Focus
Volumes
- Identify axis of rotation; washer method if hole.
- Use correct radii from sketch.
Polar coordinates
Curves r = f(θ) and area in polar form.
Cartesian link
x = r cos θ y = r sin θ r² = x² + y² Area (polar)
½ ∫_{α}^{β} r² dθ Tangent slope
dy/dx = (dy/dθ)/(dx/dθ) Topic Focus
Sketching
- Plot key θ values; note r sign and periodicity.
- Symmetry can halve the integration range.
Differential equations
First-order and simple second-order linear equations.
Separable
dy/dx = f(x) g(y) ⇒ ∫ dy/g(y) = ∫ f(x) dx Integrating factor (linear first-order)
y′ + P(x)y = Q(x); μ = exp(∫ P dx) SHM
d²x/dt² + ω² x = 0; x = A cos(ωt + φ) Auxiliary equation (2nd order)
am² + bm + c = 0; real distinct, repeated, or complex roots.
Topic Focus
Initial conditions
- Substitute to fix constants after general solution.
- Check dimensions in physics-linked DEs.
Further vectors & 3D geometry
Vector product, lines, planes, and distances.
Vector (cross) product
a × b = |a| |b| sin θ n̂ Scalar triple product
a · (b × c) — volume of parallelepiped Line
r = a + λ d Plane
r · n = p · n ax + by + cz = d Shortest distance skew lines
Use (n₁ × n₂) · (a₂ − a₁) / |n₁ × n₂| with appropriate line directions.
Topic Focus
Angles
- Line–plane angle: sin φ = |d · n| / (|d| |n|).
- Two planes: angle between normals.
Maclaurin & Taylor series
Series expansions and error bounds.
Maclaurin
f(x) = f(0) + f′(0)x + f″(0)x²/2! + … Standard expansions
e^x = 1 + x + x²/2! + … ln(1+x) = x − x²/2 + x³/3 − … sin x = x − x³/3! + … Radius of convergence
Ratio test for power series coefficients.
Topic Focus
Combination
- Substitute into known series to build new ones.
- Watch interval of validity for ln(1+x).
Further mechanics (FM options)
Useful relationships when taking Further Mechanics papers.
Work–energy
Work = ∫ F dx = Δ kinetic energy Hooke’s law (spring)
T = kx or T = λx/l (natural length l, extension x) Circular motion
a = v²/r = rω² v = rω Centre of mass
Composite bodies: x̄ = Σ m_i x_i / Σ m_i (same for ȳ, z̄).
Oblique impact
Resolve along line of centres; impulse along normal; restitution e.
Topic Focus
Diagrams
- Draw clear diagrams with forces and velocities.
- Conservation of momentum along appropriate axis.
Further statistics (FM options)
Common distributions and inference for Further Statistics routes.
Poisson
P(X = k) = e^{−λ} λ^k / k! Geometric
P(X = k) = (1 − p)^{k−1} p χ² test
Σ (O − E)² / E; compare to critical value with appropriate df.
t-distribution
Small-sample mean inference when σ unknown.
Correlation
r = S_xy / √(S_xx S_yy) Topic Focus
Assumptions
- State independence and model conditions (e.g. Poisson rate constant).
- Goodness of fit: merge expected < 5 cells.
How to Use This Formula Sheet
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Syllabus alignment
Match each block to your current 9231 syllabus; Cambridge revises topic weightings — use the latest syllabus document.
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9709 fluency
Ensure single A-level calculus, vectors, and statistics are automatic before adding further pure layers.
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Non-calculator
Know which papers allow calculators; practise exact values, surds, and induction without machine assistance where required.
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Past papers
Cambridge questions often synthesise topics — rehearse full questions, not just isolated formulas.
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 Further Mathematics Formula Sheet 2026 free to download as a PDF?
Yes. This Tutopiya formula sheet is free to use and you can download it as a PDF from this page for offline revision. There is no payment or account required for the PDF download.
What Mathematics topics and equations does this formula sheet cover?
This page groups key Mathematics formulas in one place for revision. Cambridge International AS & A Level Further Mathematics (9231) formula sheet for 2026: further pure, mechanics, and probability & statistics beyond 9709 — with full single A-level maths baseline included. 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?
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