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The Ultimate IGCSE Maths Formula Sheet Guide

Sir Faraz Hassan

Sir Faraz Hassan

Published 12 Apr 2026 · Updated 8 Aug 2026

Table of Contents▾

    Formulae are the backbone of IGCSE Mathematics. Some are given in the exam — printed on the formula sheet inside the front cover of your paper. Others you must memorise, and if you have not committed them to memory there is simply no way to answer the question. The difference between these two categories can determine whether you gain or lose four to six marks on a single problem. This guide covers every formula you need for IGCSE Maths across both Pearson Edexcel (4MA1) and Cambridge (0580), organised by topic, with clear marking of which are given and which you must learn by heart. Bookmark this page — it is your ultimate revision companion.

    Download printable PDF

    All four formula sheets compared, plus every formula you must memorise - free PDF

    40+

    formulae used across IGCSE Maths

    ~15

    formulae you must memorise (not given)

    25-30%

    of exam marks require formula application

    Given, Memorise or Not on Spec: The Critical Difference

    Both Edexcel and Cambridge print a formula sheet inside the exam paper, but they do not give you the same formulae, and neither board gives the same sheet to both of its tiers. There are four sheets in total, and the differences between them are larger than most students expect. Cambridge is the more generous of the two on area and volume: it gives you the area of a circle, the circumference and the area of a triangle at both tiers, and Edexcel gives none of those at either. Edexcel is the only board that gives the area of a trapezium, and the Edexcel Higher sheet is the only one of the four that gives the sum of an arithmetic series. Find your own sheet in the table below and learn what is missing from it.

    FormulaEdexcel FoundationEdexcel HigherCambridge CoreCambridge Extended
    Area and perimeter
    Area of a triangle (½ × base × height)MemoriseMemoriseGivenGiven
    Area of a circle (πr²)MemoriseMemoriseGivenGiven
    Circumference of a circle (2πr)MemoriseMemoriseGivenGiven
    Area of a trapeziumGivenGivenMemoriseMemorise
    Volume and surface area
    Volume of a prismGivenGivenGivenGiven
    Volume of a cylinderGivenGivenGivenGiven
    Curved surface area of a cylinderGivenGivenGivenGiven
    Volume of a pyramidNot on specNot on specGivenGiven
    Volume of a coneNot on specGivenGivenGiven
    Curved surface area of a coneNot on specGivenGivenGiven
    Volume of a sphereNot on specGivenGivenGiven
    Surface area of a sphereNot on specGivenGivenGiven
    Algebra
    Quadratic formulaNot on specGivenNot on specGiven
    Difference of two squaresMemoriseMemoriseNot on specMemorise
    Completing the squareNot on specMemoriseNot on specMemorise
    nth term of an arithmetic sequenceMemoriseMemoriseMemoriseMemorise
    Sum of an arithmetic seriesNot on specGivenNot on specNot on spec
    Trigonometry
    Pythagoras' theoremMemoriseMemoriseMemoriseMemorise
    SOH CAH TOAMemoriseMemoriseMemoriseMemorise
    Sine ruleNot on specGivenNot on specGiven
    Cosine ruleNot on specGivenNot on specGiven
    Area of a triangle (½ab sin C)Not on specGivenNot on specGiven
    Coordinate geometry
    Gradient of a lineMemoriseMemoriseMemoriseMemorise
    Equation of a line (y = mx + c)MemoriseMemoriseMemoriseMemorise
    Midpoint of a line segmentMemoriseMemoriseNot on specMemorise
    Distance between two pointsNot on specMemoriseNot on specMemorise
    Statistics, probability and finance
    MeanMemoriseMemoriseMemoriseMemorise
    Probability of an eventMemoriseMemoriseMemoriseMemorise
    Relative frequencyMemoriseMemoriseMemoriseMemorise
    Compound interestMemoriseMemoriseMemoriseMemorise
    ⚠️
    Four formulae change status between the boards. The area of a trapezium is given on Edexcel and must be memorised for Cambridge. The area of a circle, the circumference and the area of a triangle are given on Cambridge and must be memorised for Edexcel. Separately, the volume of a pyramid is on both Cambridge sheets and on neither Edexcel tier, so an Edexcel student never meets it at all. Everything else that differs, differs by tier rather than by board. And note what no sheet gives you anywhere: Pythagoras' theorem, the trigonometric ratios, every coordinate-geometry formula, and compound interest. Most of those you memorise wherever you sit; the midpoint and the distance between two points are the exception, since neither is assessed at every tier. The table says which apply to you.

    Algebra Formulae

    Algebra is the largest topic area in IGCSE Maths and these formulae underpin roughly 30 to 40 percent of the total marks across all papers. Mastering when and how to apply each one is essential for any grade above a C or Grade 5.

    1

    Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a

    Given on the Edexcel Higher and Cambridge Extended sheets, and not assessed at Edexcel Foundation or Cambridge Core. Used when you cannot factorise a quadratic equation. Identify a, b, and c from ax² + bx + c = 0, then substitute carefully. The most common error is getting the sign of b wrong or forgetting to square b before subtracting 4ac. Always simplify the discriminant (b² − 4ac) first as a separate step, then deal with the ± and the division.

    2

    Difference of two squares: a² − b² = (a + b)(a − b)

    Not given on any of the four sheets, so memorise it. Cambridge Core does not assess it at all. Recognise the pattern: two perfect squares separated by a minus sign. For example, x² − 49 = (x + 7)(x − 7). It also appears in disguised forms such as 4x² − 9 = (2x + 3)(2x − 3). Commonly tested in "factorise fully" and "simplify" questions.

    3

    Completing the square: x² + bx = (x + b/2)² − (b/2)²

    Not given on any sheet, so memorise the method if you sit Edexcel Higher or Cambridge Extended. It is not assessed at Edexcel Foundation or Cambridge Core. Used to find the turning point of a quadratic, to solve equations that do not factorise neatly, or to prove algebraic results. The turning point of y = (x + p)² + q is at (−p, q). This is a Grade 7–9 skill that appears on almost every Higher/Extended paper.

    4

    nth term of arithmetic sequence: a + (n − 1)d

    Not given — memorise. Here a is the first term, d is the common difference, and n is the position number. Typical questions ask you to find the 50th term or to determine which term equals a given value. To find n, rearrange to n = (term − a)/d + 1.

    5

    Sum of arithmetic series: S = n/2 × (2a + (n − 1)d)

    Given on the Edexcel Higher sheet - it is the first formula printed on it. Here a is the first term, d is the common difference and n is the number of terms. Cambridge 0580 does not assess the sum of an arithmetic series at either tier; it belongs to Additional Mathematics (0606). So this is an Edexcel Higher formula, given to you, and one to recognise rather than memorise.

    Quadratic Formula
    x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
    Difference of Two Squares
    a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b)
    Completing the Square (when a = 1)
    x2+bx=(x+b2)2−(b2)2x^2 + bx = \left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2
    Completing the Square (with constant c)
    x2+bx+c=(x+b2)2−(b2)2+cx^2 + bx + c = \left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2 + c
    Completing the Square (when a ≠ 1)
    ax2+bx+c=a(x+b2a)2+(c−b24a)ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right)
    nth Term of Arithmetic Sequence
    an=a+(n−1)da_n = a + (n-1)d
    Sum of Arithmetic Series
    Sn=n2(2a+(n−1)d)S_n = \frac{n}{2}\bigl(2a + (n-1)d\bigr)

    Geometry and Mensuration Formulae

    This is the most formula-heavy section of the exam, covering area, volume, and surface area. Some of these are given on the formula sheet and some are not — knowing the difference for your specific board is critical.

    1

    Area of triangle: ½ × base × height

    Given on both Cambridge sheets, NOT given on either Edexcel sheet, so memorise it if you sit Edexcel. The height must be perpendicular to the base, not the slant side. This catches students when the triangle is not right-angled: you must either drop a perpendicular from the vertex to the base, or use the alternative formula ½ab sin C if you have two sides and the included angle.

    2

    Area of trapezium: ½(a + b) × h

    Given on Edexcel, NOT given on Cambridge — memorise if sitting Cambridge. Here a and b are the two parallel sides and h is the perpendicular distance between them. The most common error is using the slant side instead of the perpendicular height.

    3

    Area of circle: πr²

    Given on both Cambridge sheets, NOT given on either Edexcel sheet, so memorise it if you sit Edexcel. Carefully distinguish this from the circumference formula (2πr or πd). The most frequent mistake is using the diameter when the formula requires the radius, or confusing area and circumference entirely.

    4

    Circumference of a circle: C = 2πr or πd

    Given on both Cambridge sheets, NOT given on either Edexcel sheet, so memorise it if you sit Edexcel. The commonest error is using the radius where the formula wants the diameter, or reaching for πr² when the question asks for the distance round the edge rather than the space inside.

    5

    Volume of cylinder: πr²h

    Given on both sheets. Understand it conceptually as area of circular cross-section multiplied by height. This principle extends to any prism: volume equals the area of the cross-section multiplied by the length. A cylinder is simply a circular prism.

    6

    Volume of a pyramid: V = ⅓ × base area × height

    Given on both Cambridge sheets. It is not on the Edexcel specification at all, at either tier, so Edexcel students can skip it. The one-third is the same one-third as the cone: a pyramid is exactly a third of the prism that encloses it, whatever the shape of its base. A cone is a pyramid with a circular base.

    7

    Volume of cone: ⅓πr²h

    Given on the Edexcel Higher, Cambridge Core and Cambridge Extended sheets. Cones are not on the Edexcel Foundation specification at all. Note the factor of one-third — a cone is exactly one-third of a cylinder with the same base radius and height. The curved surface area πrl is given on those same three sheets, where l is the slant height. If you are given r and h but not l, find it using Pythagoras: l = √(r² + h²).

    8

    Volume of sphere: ⁴⁄₃πr³

    Given on the Edexcel Higher, Cambridge Core and Cambridge Extended sheets, with the surface area 4πr² on those same three. Spheres are not on the Edexcel Foundation specification at all. A common exam question: "A hemisphere has radius 6 cm. Find its total surface area." The total surface area is half the sphere (2πr²) plus the flat circular base (πr²), giving 3πr².

    9

    Pythagoras' theorem: a² + b² = c²

    NOT given on either board — memorise. Here c is always the hypotenuse, the longest side opposite the right angle. To find a shorter side, rearrange to a² = c² − b². In three dimensions, use Pythagoras twice: first to find the diagonal of a face, then to find the space diagonal of a cuboid.

    Area of Triangle
    A=12×base×heightA = \frac{1}{2} \times \text{base} \times \text{height}
    Area of Trapezium
    A=12(a+b)×hA = \frac{1}{2}(a+b) \times h
    Area of Circle
    A=πr2A = \pi r^2
    Circumference of a Circle
    C=2πrC=πdC = 2\pi r \qquad C = \pi d
    Volume of Cylinder
    V=πr2hV = \pi r^2 h
    Volume of a Pyramid
    V=13AhV = \frac{1}{3}Ah
    Volume of Cone
    V=13πr2hV = \frac{1}{3}\pi r^2 h
    Volume & Surface Area of Sphere
    V=43πr3SA=4πr2V = \frac{4}{3}\pi r^3 \qquad SA = 4\pi r^2
    Pythagoras' Theorem
    a2+b2=c2a^2 + b^2 = c^2
    The prism shortcut
    You do not need to memorise a separate volume formula for every three-dimensional shape. For any prism — a solid with a uniform cross-section — the volume equals the area of the cross-section multiplied by the length. A cylinder is a circular prism. A triangular prism is a triangle extruded along its length. Learn the cross-section area formulae and you automatically know the volume formulae for every prism.

    Trigonometry Formulae

    Trigonometry is worth fifteen to twenty marks across your papers, split between right-angled triangle trigonometry (SOH CAH TOA) and non-right-angled triangle trigonometry (the sine rule and cosine rule). Both types appear on every Higher/Extended paper.

    1

    SOH CAH TOA

    Not given — memorise. Sin = Opposite / Hypotenuse, Cos = Adjacent / Hypotenuse, Tan = Opposite / Adjacent. Always label the sides relative to the specific angle you are working with. The hypotenuse is opposite the right angle and is always the longest side. Use the cover-up triangle method: cover the quantity you want to find, and what remains tells you whether to multiply or divide.

    2

    Sine rule: a / sin A = b / sin B = c / sin C

    Given on the Edexcel Higher and Cambridge Extended sheets, and not assessed at Edexcel Foundation or Cambridge Core. Used when you have a matching pair — a side and its opposite angle — plus one additional piece of information. The formula can also be written as sin A / a = sin B / b. Use this inverted form when you are finding an angle rather than a side. At Grade 8–9, be aware of the ambiguous case where two different triangles are possible.

    3

    Cosine rule: a² = b² + c² − 2bc cos A

    Given on the Edexcel Higher and Cambridge Extended sheets, and not assessed at Edexcel Foundation or Cambridge Core. Used when you have all three sides and need to find an angle, or when you have two sides and the included angle and need the third side. To find an angle, rearrange to cos A = (b² + c² − a²) / (2bc). This formula involves more substitution than the sine rule — practise it carefully to avoid sign errors.

    4

    Area of triangle: ½ab sin C

    Given on the Edexcel Higher and Cambridge Extended sheets, and not assessed at Edexcel Foundation or Cambridge Core. Used when you have two sides and the included angle but not the perpendicular height. C must be the angle between sides a and b — not any angle in the triangle. This is distinct from ½ × base × height, which requires the perpendicular height.

    SOH CAH TOA
    sin⁡θ=OppHypcos⁡θ=AdjHyptan⁡θ=OppAdj\sin\theta = \frac{\text{Opp}}{\text{Hyp}} \qquad \cos\theta = \frac{\text{Adj}}{\text{Hyp}} \qquad \tan\theta = \frac{\text{Opp}}{\text{Adj}}
    Sine Rule
    asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
    Cosine Rule
    a2=b2+c2−2bccos⁡Aa^2 = b^2 + c^2 - 2bc\cos A
    Area of Triangle (Trig)
    A=12absin⁡CA = \frac{1}{2}ab\sin C
    Do not confuse the two triangle area formulae
    Use ½ × base × height when you know the perpendicular height. Use ½ab sin C when you know two sides and the included angle but not the height. Using the wrong formula is a guaranteed zero on that question. Read the given information carefully: does the question provide a height or an angle? That single distinction determines which formula to apply.

    Statistics and Probability Formulae

    There are fewer formulae in this section, but they appear on every paper. Most are not given — you must memorise them.

    1

    Mean = sum of all values ÷ number of values

    Not given — memorise. For grouped data in a frequency table, the estimated mean is calculated as Σ(f × x) / Σf, where f is the frequency and x is the midpoint of each class interval. The estimated mean from a grouped frequency table is one of the most commonly examined questions at Grade 5–7 level.

    2

    Probability: P(A) = favourable outcomes / total outcomes

    Not given — memorise. For combined events with independent outcomes: P(A and B) = P(A) × P(B). For mutually exclusive events: P(A or B) = P(A) + P(B). On tree diagrams, multiply along branches to find the probability of a specific path, and add between branches to combine paths.

    3

    Relative frequency = number of successes / number of trials

    Not given — memorise. Used in experimental probability questions. As the number of trials increases, the relative frequency approaches the theoretical probability. This concept is frequently tested as an "explain why" question worth one or two marks — students must reference the law of large numbers or increasing accuracy with more trials.

    Mean
    xˉ=∑xn\bar{x} = \frac{\sum x}{n}
    Probability
    P(A)=favourable outcomestotal outcomesP(A) = \frac{\text{favourable outcomes}}{\text{total outcomes}}

    Coordinate Geometry Formulae

    None of these formulae are given on any of the four sheets, so whatever your tier assesses, you memorise. The gradient and the equation of a line are assessed on all four. The midpoint is not assessed at Cambridge Core, and the distance between two points only at Edexcel Higher and Cambridge Extended. They appear in almost every Paper 2 (Edexcel) and Paper 4 (Cambridge).

    1

    Gradient: m = (y₂ − y₁) / (x₂ − x₁)

    Not given — memorise. Rise over run. A positive gradient slopes upward from left to right; a negative gradient slopes downward. Parallel lines have equal gradients. Perpendicular lines have gradients that multiply to −1 — they are negative reciprocals of each other.

    2

    Equation of a line: y = mx + c or y − y₁ = m(x − x₁)

    Not given — memorise both forms. Use y = mx + c when you know the gradient and the y-intercept. Use y − y₁ = m(x − x₁) when you know the gradient and any point on the line. The second form is more versatile and I recommend learning it thoroughly — it works in every situation.

    3

    Midpoint: ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

    Not given on any sheet, so memorise it. Cambridge Core does not assess it. This is simply the average of the x-coordinates and the average of the y-coordinates. Used in questions about finding the centre of a line segment, proving properties of quadrilaterals, and coordinate geometry proofs.

    4

    Distance: d = √((x₂ − x₁)² + (y₂ − y₁)²)

    Not given on any sheet, so memorise it if you sit Edexcel Higher or Cambridge Extended. It is not assessed at Edexcel Foundation or Cambridge Core. This is Pythagoras' theorem applied to coordinates. The horizontal distance is (x₂ − x₁), the vertical distance is (y₂ − y₁), and the actual distance between the two points is the hypotenuse of the right-angled triangle formed by these two components.

    Gradient
    m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}
    Equation of a Line
    y−y1=m(x−x1)y - y_1 = m(x - x_1)
    Midpoint
    M=(x1+x22,  y1+y22)M = \left(\frac{x_1 + x_2}{2},\; \frac{y_1 + y_2}{2}\right)
    Distance
    d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

    How to Actually Memorise These Formulae

    Knowing which formulae to memorise is half the battle. Actually committing them to long-term memory is the other half. Here are the techniques that work — grounded in cognitive science and tested with hundreds of my own students.

    1

    Write them out daily — by hand

    Every morning for three weeks, write out all fifteen memorise-only formulae from memory on a blank piece of paper. No peeking. Check against this guide afterwards and mark which ones you missed. The next morning, start with the ones you got wrong. By day fourteen, you will write them all in under five minutes without thinking. Handwriting engages motor memory in a way that typing does not — use pen and paper for this exercise.

    2

    Use them in context — not in isolation

    Do not just memorise “a² + b² = c²” as an abstract string. Immediately after writing the formula, solve five Pythagoras questions. Your brain stores the formula alongside the context of using it. When you see a right-angled triangle in the exam, the formula surfaces automatically because you have paired the formula with the visual cue in practice.

    3

    Flashcards with a twist

    Put the question type on the front of the card: “Find the area of a triangle given two sides and an included angle.” Put the formula on the back: ½ab sin C. This is more effective than putting the formula name on the front, because in the exam you start with the question, not the formula name. Train your brain to retrieve the formula from the question context, not the other way around.

    4

    The exam-morning ritual

    On the morning of your exam, before you enter the hall, write every memorised formula on a piece of scrap paper from memory. Check it against your notes. This is your final confidence boost. When the exam begins, immediately write all formulae on the inside cover of your answer booklet before reading Question 1. Now they are on the page in front of you and cannot be forgotten under pressure.

    Formula Mastery Checklist

    • I know which formulae are given and which I must memorise for my exam board
    • I can write all memorise-only formulae from memory in under 5 minutes
    • I can identify which formula to use from the question context alone
    • I know the difference between ½bh and ½ab sin C for triangle area
    • I can rearrange the cosine rule to find an angle
    • I know the gradient and midpoint formulae without hesitation
    • I can apply Pythagoras' theorem in both 2D and 3D problems
    • I understand that volume of any prism = cross-section area × length
    • I practise writing formulae from memory every morning
    • I use context-based flashcards, not formula-name flashcards

    The students who ace the formula questions are not the ones with the best memories — they are the ones who practised writing them every day until it became automatic. Formulae are like phone numbers used to be: write it enough times and you will never forget it.

    Sir Faraz Hassan — GCSE & IGCSE Maths Specialist

    Download printable PDF

    Print it, stick it above your desk, and write the memorise list out daily.

    Frequently Asked Questions

    Yes, but there are four different sheets, not two. Pearson Edexcel (4MA1) and Cambridge (0580) each print one inside the front cover, and each board gives a different sheet to each of its tiers. Edexcel Foundation gives four formulae, Edexcel Higher gives thirteen, Cambridge Core gives eleven and Cambridge Extended gives fifteen. Some formulae appear on no sheet at all, including Pythagoras' theorem, the trigonometric ratios (SOH CAH TOA) and compound interest.

    Yes, and there are two of them. The Foundation sheet gives just four formulae: the area of a trapezium, the volume of a prism, the volume of a cylinder and the curved surface area of a cylinder. The Higher sheet gives thirteen, adding the quadratic formula, the sine and cosine rules, the area of a triangle from two sides and the included angle, the sum of an arithmetic series, and the cone and sphere formulae. Whichever tier you sit, you must memorise Pythagoras' theorem, SOH CAH TOA, the area of a circle, the circumference, compound interest, the gradient of a line, its equation and the midpoint of a line segment. Higher students must also know how to find the distance between two points, which is not assessed at Foundation.

    Yes, and Cambridge is the more generous of the two boards on area and volume. The Core sheet gives eleven formulae and the Extended sheet gives fifteen. Both include the area of a circle, the circumference and the area of a triangle, none of which Edexcel gives at either tier. Extended adds the quadratic formula, the sine and cosine rules, and the area of a triangle from two sides and the included angle. What Cambridge does not give, and Edexcel does, is the area of a trapezium.

    About fifteen, though the exact figure depends on which of the four sheets you sit: Edexcel Foundation leaves fourteen to memorise, Edexcel Higher sixteen, Cambridge Core ten and Cambridge Extended fourteen. Nine are missing from every sheet: Pythagoras' theorem, SOH CAH TOA, the gradient of a line, the equation of a straight line, the nth term of a sequence, the mean, basic probability, relative frequency and compound interest. Edexcel students must also memorise the area of a circle, the circumference and the area of a triangle. Cambridge students must memorise the area of a trapezium.

    No, and there are four sheets rather than two, because each board gives a different one to each tier. Four formulae change status between the boards: the area of a trapezium is given on Edexcel and must be memorised for Cambridge, while the area of a circle, the circumference and the area of a triangle are given on Cambridge and must be memorised for Edexcel. Separately, the volume of a pyramid is on both Cambridge sheets and on neither Edexcel tier, so an Edexcel student never meets it at all. Everything else that differs between the four sheets differs by tier rather than by board. If you revise from another board's resources, check which formulae change status rather than assuming they are identical.

    There are more than 40 formulae across IGCSE Maths, of which roughly 15 must be memorised. Formula application accounts for about 25–30% of the total exam marks, so knowing them accurately — and recognising which question each one applies to — can be worth a full grade or more.

    Past papers

    Try these formulae on real papers

    Seventeen Edexcel 4MA1 papers, every question with a worked solution and an M1/A1 mark scheme.

    Past Paper Solutions

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