igcse tips14 min read

The Ultimate IGCSE Maths Formula Sheet Guide

Sir Faraz Hassan

12 Apr 2026

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.

40+

formulae used across IGCSE Maths

~15

formulae you must memorise (not given)

25-30%

of exam marks require formula application

Given vs Must Memorise: The Critical Difference

Both Edexcel and Cambridge provide a formula sheet inside the exam paper, but they do not give you the same formulae. Edexcel's sheet is more generous; Cambridge expects you to memorise more. This is a genuine strategic difference between the boards. If you are sitting Cambridge, you need to commit more to memory. If you are on Edexcel, you still need to understand how to use the given formulae — having a formula on the sheet is useless if you cannot recognise which question it applies to.

Formula	Edexcel (4MA1)	Cambridge (0580)
Quadratic formula	Given	Given
Area of trapezium	Given	NOT given — memorise
Volume of prism	Given	NOT given — memorise
Volume of cylinder	Given	Given
Volume of cone	Given	Given
Volume of sphere	Given	Given
Surface area of sphere	Given	Given
Curved SA of cone	Given	Given
Pythagoras' theorem	NOT given — memorise	NOT given — memorise
Trig ratios (SOH CAH TOA)	NOT given — memorise	NOT given — memorise
Sine rule	Given	Given
Cosine rule	Given	Given
Area of triangle (½ab sin C)	Given	Given
Compound interest	Given	NOT given — memorise
Gradient formula	NOT given — memorise	NOT given — memorise
Distance between points	NOT given — memorise	NOT given — memorise
Midpoint formula	NOT given — memorise	NOT given — memorise

⚠️

Cambridge students: you must memorise the area of a trapezium, volume of a prism, and compound interest formula. Edexcel gives all three on the formula sheet. If you are switching boards or using resources from the other board, always verify which formulae change status — do not assume they are the same.

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.

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

Given on both formula sheets. 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.

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

Not given on either sheet — memorise. 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.

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

Not given — memorise the method. 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.

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.

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

Cambridge Extended only — not tested on Edexcel 4MA1. Not given — memorise if you sit Cambridge. Used for questions such as "find the sum of the first 20 terms of the sequence 3, 7, 11, 15, …". Identify a = 3, d = 4, n = 20, then substitute.

Quadratic Formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Difference of Two Squares

a^2 - b^2 = (a+b)(a-b)

Completing the Square (when a = 1)

x^2 + bx = \left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2

Completing the Square (with constant c)

x^2 + bx + c = \left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2 + c

Completing the Square (when a ≠ 1)

ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right)

nth Term of Arithmetic Sequence

a_n = a + (n-1)d

Sum of Arithmetic Series

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.

Area of triangle: ½ × base × height

Not given — memorise. 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.

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.

Area of circle: πr²

Not given — memorise. 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.

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.

Volume of cone: ⅓πr²h

Given on both sheets. 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 also given, where l is the slant height. If you are given r and h but not l, find it using Pythagoras: l = √(r² + h²).

Volume of sphere: ⁴⁄₃πr³

Given on both sheets. Surface area 4πr² is also given. 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.

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 = \frac{1}{2} \times \text{base} \times \text{height}

Area of Trapezium

A = \frac{1}{2}(a+b) \times h

Area of Circle

A = \pi r^2

Volume of Cylinder

V = \pi r^2 h

Volume of Cone

V = \frac{1}{3}\pi r^2 h

Volume & Surface Area of Sphere

V = \frac{4}{3}\pi r^3 \qquad SA = 4\pi r^2

Pythagoras' Theorem

a^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.

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.

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

Given on both formula sheets. 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.

Cosine rule: a² = b² + c² − 2bc cos A

Given on both formula sheets. 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.

Area of triangle: ½ab sin C

Given on both formula sheets. 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\theta = \frac{\text{Opp}}{\text{Hyp}} \qquad \cos\theta = \frac{\text{Adj}}{\text{Hyp}} \qquad \tan\theta = \frac{\text{Opp}}{\text{Adj}}

Sine Rule

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Cosine Rule

a^2 = b^2 + c^2 - 2bc\cos A

Area of Triangle (Trig)

A = \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.

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.

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.

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

\bar{x} = \frac{\sum x}{n}

Probability

P(A) = \frac{\text{favourable outcomes}}{\text{total outcomes}}

Coordinate Geometry Formulae

None of these formulae are given on the formula sheet for either board. All must be memorised. They appear in almost every Paper 2 (Edexcel) and Paper 4 (Cambridge).

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.

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.

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

Not given — memorise. 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.

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

Not given — memorise. 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 = \frac{y_2 - y_1}{x_2 - x_1}

Equation of a Line

y - y_1 = m(x - x_1)

Midpoint

M = \left(\frac{x_1 + x_2}{2},\; \frac{y_1 + y_2}{2}\right)

Distance

d = \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.

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.

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.

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.

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

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