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AQA A-level Maths formula booklet

AQA gives you a formulae booklet in every A-level Maths exam, and a bigger one with statistical tables for Further Maths. Below, we list every formula in our notes from a topic on the AQA specification and say whether the booklet prints it or AQA says you must know it.

167 formulae · 48 printed · 47 must know · 72 to check.

Where to find it

Both booklets are PDFs on AQA’s website: Formulae for A-level Mathematics (7356, 7357) and Formulae and statistical tables for Maths and Further Maths (7366, 7367). Both are version 2.0, issued June 2025. You can use the bigger one in a Maths exam too.

The formulae you must be able to use without being given them are listed in Appendix B of each specification: A-level Mathematics (7357), page 45, and A-level Further Mathematics (7367), page 51.

Must know or printed: the difference

Must know means the formula is on Appendix B. AQA expects you to use it without being given it, so it is not in the booklet and you need to learn it. Printed means the booklet on your desk has it, so the job is knowing where it is and how to use it, not memorising it. The two lists do not cover everything. A formula on neither is marked check: the booklet will not give it to you either, so learn it or know how to derive it.

What the marks mean

Printed: the booklet has it, maybe rearranged, with different letters, or in a form you can read it straight off. Must know: Appendix B lists it, in this form or an equivalent one. Check: it is on neither list in this form, and the line under it says why. A formula from a Maths topic is checked against the Maths booklet, the one you are given in a Maths exam; one from a Further Maths topic only is checked against the bigger booklet. The line under each formula names the booklet or the appendix.

The formulae in these notes, by unit

Every formula displayed in a lesson that covers the AQA specification, linked to its unit’s formulae page, where it is written out with its conditions.

ALGEBRA AND FUNCTIONS

• a^x × a^y = a^x+yMUST KNOWIndices and surdsAppendix B, 7357
• a^x ÷ a^y = a^x−yMUST KNOWIndices and surdsAppendix B, 7357
• (a^x)^y = a^xyMUST KNOWIndices and surdsAppendix B, 7357
• √xy = √x√yCHECKIndices and surdsAQA neither prints it nor lists it in Appendix B
• ax² + bx + c = a(x + b2a)² + (c − b²4a)CHECKQuadratic functionsA working line: completing the square; neither AQA list has it
• x = −b ± √b² − 4ac2aMUST KNOWQuadratic functionsAppendix B, 7357
• f(a) = 0 ⇔ (x − a) is a factor of f(x)CHECKPolynomials and the factor theoremAQA neither prints it nor lists it in Appendix B

COORDINATE GEOMETRY

• y − y₁ = m(x − x₁)MUST KNOWStraight linesAppendix B, 7357
• m₁ m₂ = −1MUST KNOWStraight linesAppendix B, 7357
• (x − a)² + (y − b)² = r²CHECKCirclesAQA neither prints it nor lists it in Appendix B

SEQUENCES AND SERIES

• ⁿC_r = n!r!(n − r)!PRINTEDThe binomial expansionMaths booklet
• (a + b)ⁿ = aⁿ + ⁿC₁aⁿ⁻¹b + … + bⁿPRINTEDThe binomial expansionMaths booklet
• (1 + x)ⁿ = 1 + nx + n(n − 1)2!x² + …PRINTEDThe general binomial expansionMaths booklet
• (a + bx)ⁿ = aⁿ(1 + bx/a)ⁿCHECKThe general binomial expansionA working line: AQA prints (1 + x)ⁿ, and this step takes aⁿ out to reach it
• u_n = a + (n − 1)dMUST KNOWArithmetic seriesAppendix B, 7357
• S_n = n2(2a + (n − 1)d) = n2(a + l)PRINTEDArithmetic seriesMaths booklet
• u_n = arⁿ⁻¹MUST KNOWGeometric seriesAppendix B, 7357
• S_n = a(1 − rⁿ)1 − rPRINTEDGeometric seriesMaths booklet
• S_∞ = a1 − rPRINTEDGeometric seriesMaths booklet

TRIGONOMETRY

• asin A = bsin B = csin CMUST KNOWTriangles and the sine and cosine rulesAppendix B, 7357
• a² = b² + c² − 2bc cos AMUST KNOWTriangles and the sine and cosine rulesAppendix B, 7357
• Area = 12ab sin CMUST KNOWTriangles and the sine and cosine rulesAppendix B, 7357
• tan θ = sin θcos θCHECKTrigonometric graphs and equationsAQA neither prints it nor lists it in Appendix B
• sin ² θ + cos ² θ = 1MUST KNOWTrigonometric graphs and equationsAppendix B, 7357
• s = rθMUST KNOWRadians, arcs and small anglesAppendix B, 7357
• A = 12r²θMUST KNOWRadians, arcs and small anglesAppendix B, 7357
• sin θ ≈ θ, cos θ ≈ 1 − θ²2, tan θ ≈ θPRINTEDRadians, arcs and small anglesMaths booklet
• sec² θ = 1 + tan² θMUST KNOWReciprocal and inverse trigonometric functionsAppendix B, 7357
• cosec² θ = 1 + cot² θMUST KNOWReciprocal and inverse trigonometric functionsAppendix B, 7357
• sin (A ± B) = sin A cos B ± cos A sin BPRINTEDCompound angles and the harmonic formMaths booklet
• cos (A ± B) = cos A cos B ∓ sin A sin BPRINTEDCompound angles and the harmonic formMaths booklet
• tan (A ± B) = tan A ± tan B1 ∓ tan A tan BPRINTEDCompound angles and the harmonic formMaths booklet
• sin 2A = 2 sin A cos AMUST KNOWCompound angles and the harmonic formAppendix B, 7357
• cos 2A = cos ²A − sin ²A = 2cos ²A − 1 = 1 − 2sin ²AMUST KNOWCompound angles and the harmonic formAppendix B, 7357
• tan 2A = 2tan A1 − tan ²AMUST KNOWCompound angles and the harmonic formAppendix B, 7357

EXPONENTIALS AND LOGARITHMS

• gradient of e^kx = ke^kxMUST KNOWExponential functions and eAppendix B, 7357
• x = aⁿ ⇔ n = log_a xMUST KNOWLogarithms and their lawsAppendix B, 7357
• log_a x + log_a y = log_a xyMUST KNOWLogarithms and their lawsAppendix B, 7357
• log_a x − log_a y = log_a xyMUST KNOWLogarithms and their lawsAppendix B, 7357
• k log_a x = log_a x^kMUST KNOWLogarithms and their lawsAppendix B, 7357

DIFFERENTIATION

• f′(x) = lim_{h → 0} f(x + h) − f(x)hPRINTEDThe derivative from first principlesMaths booklet
• y = xⁿ ⇒ dydx = nxⁿ⁻¹MUST KNOWDifferentiating powers of xAppendix B, 7357
• sin kx → kcos kx, cos kx → −ksin kxMUST KNOWDifferentiating trig, exponentials and logsAppendix B, 7357
• e^kx → ke^kx, ln x → 1xMUST KNOWDifferentiating trig, exponentials and logsAppendix B, 7357
• dydx = dydu × dudxMUST KNOWThe product, quotient and chain rulesAppendix B, 7357
• (uv)′ = u′v − uv′v²PRINTEDThe product, quotient and chain rulesMaths booklet
• dydx = 1dx/dy = 1cos yCHECKImplicit and parametric differentiationA working line: differentiating x as a function of y and inverting; neither AQA list has it

INTEGRATION

• ∫ xⁿ dx = xⁿ⁺¹n + 1 + c (n ≠ −1)MUST KNOWIntegration as antidifferentiationAppendix B, 7357
• ∫ cos kx dx = 1ksin kx + cMUST KNOWIntegrating standard functionsAppendix B, 7357
• ∫ sin kx dx = −1kcos kx + cMUST KNOWIntegrating standard functionsAppendix B, 7357
• ∫ e^kx dx = 1ke^kx + cMUST KNOWIntegrating standard functionsAppendix B, 7357
• ∫ 1x dx = ln|x| + cMUST KNOWIntegrating standard functionsAppendix B, 7357
• ∫ f′(x)f(x) dx = ln|f(x)| + cPRINTEDIntegration by substitution and by partsMaths booklet
• ∫ u dvdx dx = uv − ∫ v dudx dxPRINTEDIntegration by substitution and by partsMaths booklet
• ∫ 23x + 5 dx = 23ln|3x + 5| + cCHECKIntegrating rational functionsA worked case of ∫f′(x)/f(x) dx = ln|f(x)| + c, which AQA prints
• area = ∫ y dxdt dtCHECKAreas, parametric curves and the limit of a sumNot printed; Appendix B lists the area under a curve, ∫y dx, and this is that integral with x = x(t) substituted
• lim_{δx → 0} Σ f(x) δx = ∫_a^b f(x) dxCHECKAreas, parametric curves and the limit of a sumA working line: the definite integral as the limit of a sum; neither AQA list has it
• ∫ 1g(y) dy = ∫ f(x) dxCHECKSolving differential equationsA working line: separating the variables; neither AQA list has it

NUMERICAL METHODS

• x_n+1 = (x_n + 1)^1/3CHECKLocating roots and iterationA working line: a worked iteration; neither AQA list has it
• x_n+1 = x_n − f(x_n)f′(x_n)PRINTEDThe Newton-Raphson methodMaths booklet
• T = h2(y₀ + 2(y₁ + … + y_n−1) + y_n)PRINTEDThe trapezium ruleMaths booklet
• L = h(y₀ + … + y_n−1),     R = h(y₁ + … + y_n)CHECKThe trapezium ruleA working line: rectangle sums beside the trapezium rule; neither AQA list has it

VECTORS

• AB = b − aCHECKVectors in two dimensionsAQA neither prints it nor lists it in Appendix B
• |xi + yj + zk| = √x² + y² + z²MUST KNOWVectors in three dimensionsAppendix B, 7357

STATISTICS

• stratum sizepopulation size × sample sizeCHECKSampling and the large data setAQA neither prints it nor lists it in Appendix B
• mean: Σ xnMUST KNOWMeasures of location and spreadAppendix B, 7357
• σ² = Σ x²n − (Σ xn)²PRINTEDMeasures of location and spreadMaths booklet
• mean(x) = a + b mean(y),     σ_x = |b|σ_yCHECKMeasures of location and spreadAQA neither prints it nor lists it in Appendix B
• frequency density = frequencyclass widthCHECKRepresenting and interpreting dataAQA neither prints it nor lists it in Appendix B
• outlier: beyond Q₁ − 1.5 × IQR or Q₃ + 1.5 × IQRCHECKRepresenting and interpreting dataAQA neither prints it nor lists it in Appendix B
• P(A ∪ B) = P(A) + P(B) − P(A ∩ B)PRINTEDProbability and Venn diagramsMaths booklet
• P(A|B) = P(A ∩ B)P(B)PRINTEDConditional probabilityMaths booklet
• P(X = x) = 1nCHECKThe binomial distributionAQA neither prints it nor lists it in Appendix B
• P(X = x) = ⁿC_x p^x (1 − p)^{n − x}PRINTEDThe binomial distributionMaths booklet
• Z = X − μσMUST KNOWThe normal distributionAppendix B, 7357
• X ~ B(n, p) ≈ N(np, np(1 − p))CHECKThe normal distributionAQA neither prints it nor lists it in Appendix B
• mean of n observations: N(μ, σ²/n)PRINTEDHypothesis testing: correlation and the normalMaths booklet

MECHANICS

• v = u + at,     s = (u + v)2tPRINTEDKinematics with constant accelerationMaths booklet
• s = ut + 12at²,     v² = u² + 2asPRINTEDKinematics with constant accelerationMaths booklet
• s = vt − 12at²PRINTEDKinematics with constant accelerationMaths booklet
• v = dxdt,     a = dvdtMUST KNOWKinematics with variable accelerationAppendix B, 7357
• F = maMUST KNOWForces and Newton's lawsAppendix B, 7357
• y = x tan θ − gx²2u²cos ² θCHECKProjectilesA working line: the path of a projectile; neither AQA list has it
• F ≤ μRMUST KNOWFriction and inclined planesAppendix B, 7357
• |m_Bg−m_Agsin θ|≤ μm_Agcos θCHECKFriction and inclined planesA working line: the lesson's test for equilibrium on a rough plane; neither AQA list has it
• moment = force × perpendicular distanceCHECKMomentsAQA neither prints it nor lists it in Appendix B

FURTHER PROOF

• true for n = 1, and true for k + 1 whenever true for kCHECKProof by induction: sums and seriesA working line: the structure of a proof by induction; neither AQA list has it

COMPLEX NUMBERS

• z = r(cos θ + i sin θ)MUST KNOWModulus, argument and lociAppendix B, 7367
• (cos θ + i sin θ)ⁿ = cos nθ + i sin nθPRINTEDDe Moivre's theorem and trigonometric identitiesFurther Maths booklet

FURTHER ALGEBRA AND SERIES

• α + β + γ = −ba, αβ + αγ + βγ = ca, αβγ = −daMUST KNOWRoots of polynomialsAppendix B, 7367
• Σ r = n(n+1)2MUST KNOWSumming series and the method of differencesAppendix B, 7367
• Σ r² = n(n+1)(2n+1)6, Σ r³ = n²(n+1)²4PRINTEDSumming series and the method of differencesFurther Maths booklet
• f(x) = f(0) + f′(0)x + f″(0)2!x² + …PRINTEDMaclaurin seriesFurther Maths booklet
• ln(1 + x) = x − x²2 + x³3 − x⁴4 + … (−1 < x ≤ 1)PRINTEDMaclaurin seriesFurther Maths booklet

FURTHER VECTORS

• r = a + λbMUST KNOWLines and planes in three dimensionsAppendix B, 7367
• r · n = a · n = dMUST KNOWLines and planes in three dimensionsAppendix B, 7367
• distance = |n₁x₀ + n₂y₀ + n₃z₀ − d||n|CHECKIntersections, angles and distancesAQA neither prints it nor lists it in Appendix B

FURTHER CALCULUS

• V = π ∫_a^b y² dxMUST KNOWVolumes of revolutionAppendix B, 7367
• mean value = 1b − a ∫_a^b f(x) dxCHECKMean values and improper integralsAQA neither prints it nor lists it in Appendix B
• ddx(arcsin x) = 1√1 − x²PRINTEDCalculus with inverse trigonometric functionsFurther Maths booklet
• ddx(arctan x) = 11 + x²PRINTEDCalculus with inverse trigonometric functionsFurther Maths booklet

HYPERBOLIC FUNCTIONS

• cosh x = e^x + e^−x2, sinh x = e^x − e^−x2MUST KNOWHyperbolic functions and identitiesAppendix B, 7367
• ∫ 1√x² + a² dx = arsinhxa + cPRINTEDCalculus with hyperbolic functionsFurther Maths booklet

POLAR COORDINATES

• x = r cos θ, y = r sin θ, r² = x² + y²CHECKPolar curvesAQA neither prints it nor lists it in Appendix B
• A = 12 ∫_α^β r² dθPRINTEDAreas with polar coordinatesFurther Maths booklet

DIFFERENTIAL EQUATIONS

• IF = e^{∫ P dx}CHECKFirst order equations and integrating factorsAQA neither prints it nor lists it in Appendix B
• am² + bm + c = 0CHECKSecond order equationsAQA neither prints it nor lists it in Appendix B

FURTHER PURE 1

• x²a² + y²b² = 1, x = a cos t, y = b sin tPRINTEDConic sectionsFurther Maths booklet
• a × b = (a₂b₃ − a₃b₂, a₃b₁ − a₁b₃, a₁b₂ − a₂b₁)PRINTEDThe vector product and the scalar triple productFurther Maths booklet
• dydx ≈ y_n+1 − y_nh or y_n+1 − y_n−12hCHECKNumerical methods for differential equationsAQA prints Euler's method and the improved Euler method, which are these two approximations rearranged into recurrences
• ∫_a^b y dx ≈ h3(y₀ + 4y₁ + 2y₂ + … + 4y_n−1 + y_n)PRINTEDNumerical methods for differential equationsFurther Maths booklet

FURTHER STATISTICS 1

• E(X) = Σ x P(X = x)CHECKDiscrete random variables and expectationAQA neither prints it nor lists it in Appendix B
• Var(X) = Σ x² P(X = x) − μ²CHECKDiscrete random variables and expectationAQA neither prints it nor lists it in Appendix B
• E(X) = n + 12,      Var(X) = n² − 112CHECKDiscrete random variables and expectationAQA neither prints it nor lists it in Appendix B
• P(X = x) = e^−λ λ^xx!, E(X) = Var(X) = λPRINTEDThe Poisson distributionFurther Maths booklet
• χ² = Σ (O_i − E_i)²E_iPRINTEDGoodness-of-fit testsFurther Maths booklet
• E_ij = row total × column totalgrand totalCHECKContingency tablesAQA neither prints it nor lists it in Appendix B
• χ² = Σ (O_i − E_i)²E_iPRINTEDContingency tablesFurther Maths booklet
• χ² = Σ (|O − E| − 0.5)²ECHECKContingency tablesAQA prints Σ(O − E)²/E without the continuity correction

FURTHER PURE 2

• n I_n = (n − 1) I_n−2, I_n = ∫₀^π/2 sin ⁿ x dxCHECKReduction formulaeA working line: a reduction formula worked in the lesson; neither AQA list has it
• s = ∫ √1 + (dydx)² dxPRINTEDArc length and surface areaFurther Maths booklet
• det(M − λI) = 0CHECKEigenvalues and eigenvectorsAQA neither prints it nor lists it in Appendix B
• P⁻¹MP = DCHECKDiagonalisation and the Cayley-Hamilton theoremAQA neither prints it nor lists it in Appendix B
• a^p−1 ≡ 1 (mod p)CHECKModular arithmetic and Fermat's little theoremAQA neither prints it nor lists it in Appendix B

FURTHER STATISTICS 2

• P(a < X ≤ b) = ∫_a^b f(x) dxCHECKContinuous random variables: density and distribution functionsAQA neither prints it nor lists it in Appendix B
• f(x) = dF(x)dxCHECKContinuous random variables: density and distribution functionsAQA neither prints it nor lists it in Appendix B
• Σ P(X = c) + ∫ f(x) dx = 1CHECKContinuous random variables: density and distribution functionsAQA neither prints it nor lists it in Appendix B
• P(X > x) = e^−λx,      F(x) = 1 − e^−λxCHECKContinuous random variables: density and distribution functionsAQA neither prints it nor lists it in Appendix B
• f(x) = λe^−λx for x ≥ 0CHECKContinuous random variables: density and distribution functionsAQA neither prints it nor lists it in Appendix B
• P(X > s + t | X > s) = P(X > t)CHECKContinuous random variables: density and distribution functionsAQA neither prints it nor lists it in Appendix B
• E(X) = ∫ x f(x) dx,      Var(X) = E(X²) − [E(X)]²CHECKMean, variance and skewness of continuous variablesAQA neither prints it nor lists it in Appendix B
• E(X) = 1λ,      Var(X) = 1λ²CHECKMean, variance and skewness of continuous variablesAQA neither prints it nor lists it in Appendix B
• F(x) = 0 for x < a,      x − ab − a for a ≤ x ≤ b,      1 for x > bCHECKThe continuous uniform distributionAQA neither prints it nor lists it in Appendix B
• E(X) = a + b2,      Var(X) = (b − a)²12CHECKThe continuous uniform distributionAQA neither prints it nor lists it in Appendix B
• aX ± bY is normal, with mean aμ_x ± bμ_yCHECKCombinations of normal random variablesAQA neither prints it nor lists it in Appendix B
• Var(aX ± bY) = a²σ_x² + b²σ_y²CHECKCombinations of normal random variablesAQA neither prints it nor lists it in Appendix B
• s² = 1n − 1 Σ (x_i − sample mean)²PRINTEDEstimators, standard error and confidence intervalsFurther Maths booklet
• s² = 1n − 1(Σ x² − (Σ x)²n)CHECKEstimators, standard error and confidence intervalsAQA prints S² as the sum of (X − X̄)² over n − 1; this computational form is that sum expanded
• sample mean ± z × σ√n,     z = 1.96 at 95 per centCHECKEstimators, standard error and confidence intervalsAQA prints (X̄ − μ)/(σ/√n) ~ N(0, 1), which the interval is built from, but not the interval itself
• sample mean − μS√n is t on n − 1 degrees of freedomPRINTEDConfidence intervals and tests with the t-distributionFurther Maths booklet
• s² = (n₁ − 1)s₁² + (n₂ − 1)s₂²n₁ + n₂ − 2CHECKConfidence intervals and tests with the t-distributionAQA prints the single-sample S² but not the pooled estimate

FURTHER MECHANICS 1

• I = Ft = mv − muCHECKMomentum and impulseAQA neither prints it nor lists it in Appendix B
• m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂CHECKMomentum and impulseAQA neither prints it nor lists it in Appendix B
• I = mv − muCHECKImpulse and momentum as vectorsAQA neither prints it nor lists it in Appendix B
• KE = 12mv²,      GPE = mghCHECKWork, energy and powerAQA neither prints it nor lists it in Appendix B
• P = FvCHECKWork, energy and powerAQA neither prints it nor lists it in Appendix B
• T = λ xlCHECKHooke's law and elastic stringsAQA neither prints it nor lists it in Appendix B
• EPE = λ x²2lCHECKElastic potential energyAQA neither prints it nor lists it in Appendix B
• e = speed of separationspeed of approachCHECKDirect impact and Newton's law of restitutionAQA neither prints it nor lists it in Appendix B

FURTHER MECHANICS 2

• a = rω² = v²rCHECKAngular speed and horizontal circular motionAQA neither prints it nor lists it in Appendix B
• tan θ = v²rgCHECKAngular speed and horizontal circular motionAQA neither prints it nor lists it in Appendix B
• T − mgcos θ = mv²rCHECKMotion in a vertical circleA working line: Newton's second law along the radius of a vertical circle; neither AQA list has it
• mean x = Σ m_ix_iΣ m_i,      mean y = Σ m_iy_iΣ m_iCHECKCentre of mass of a discrete distributionAQA neither prints it nor lists it in Appendix B
• Triangular lamina: 23 along the median from the vertexPRINTEDCentres of mass of plane figures and frameworksFurther Maths booklet
• Circular arc, radius r, angle at centre 2α: r sin αα from the centrePRINTEDCentres of mass of plane figures and frameworksFurther Maths booklet
• Sector of a circle, radius r, angle at centre 2α: 2r sin α3α from the centrePRINTEDCentres of mass of plane figures and frameworksFurther Maths booklet
• Solid hemisphere, radius r: 38r from the centrePRINTEDCentres of mass of plane figures and frameworksFurther Maths booklet
• Hemispherical shell, radius r: 12r from the centrePRINTEDCentres of mass of plane figures and frameworksFurther Maths booklet
• Solid cone or pyramid, height h: 14h above the base, on the line from the centre of the base to the vertexPRINTEDCentres of mass of plane figures and frameworksFurther Maths booklet
• Conical shell, height h: 13h above the base, on the same linePRINTEDCentres of mass of plane figures and frameworksFurther Maths booklet
• mean x = ∫ xy dx∫ y dx,      mean y = ∫ 12y² dx∫ y dxCHECKCentres of mass by integrationAQA neither prints it nor lists it in Appendix B
• mean x = ∫ x y² dx∫ y² dxCHECKCentres of mass by integrationAQA neither prints it nor lists it in Appendix B
• F(x) = mvdvdx,     F(t) = mdvdtCHECKNewton's laws with a variable forceAppendix B lists F = ma and a = dv/dt, which give the dv/dt form; neither list has the v dv/dx form
• d²xdt² = −ω²xMUST KNOWSimple harmonic motionAppendix B, 7367
• v² = ω²(a² − x²)CHECKSimple harmonic motionAQA neither prints it nor lists it in Appendix B
• md²xdt² = −λ xl,     ω² = λmlCHECKOscillations on strings and springsA working line: the equation of motion for an elastic string or spring; neither AQA list has it

DECISION MATHEMATICS 1

• 3x + 2y ≤ 20 becomes 3x + 2y + s₁ = 20CHECKLinear programming: formulation and graphical solutionA working line: a slack variable added to a constraint; neither AQA list has it

How this was checked: both booklets and both Appendix B lists were copied out line by line from AQA’s own PDFs, and every formula was matched to them by the mathematics rather than by its name. Each mark is tied to the line it was matched to, and the site will not build if that line leaves its list. If the booklet you are given in the exam ever disagrees with this page, the booklet is right.

Other boards: OCR A-level Maths formula sheet · OCR MEI A-level Maths formula sheet · Pearson Edexcel, on the formulae index