IGCSE Maths Formula Sheet: What You Actually Need

If you searched for an IGCSE Maths formula sheet, you probably hope for a one-page printout you can take into the exam. That is the first thing to clear up: Cambridge IGCSE Mathematics (syllabus 0580) does not provide a formula sheet in the exam. Edexcel IGCSE Mathematics A (4MA1) does not provide one either. You walk into the exam hall with your calculator, your pens, and whatever you have memorised.

This catches students off guard every year. In the IB Diploma you get a substantial formula booklet for Mathematics. In A-Level you get a printed list of statistical formulas, integrals and identities. In IGCSE you get nothing.

That makes the question of which formulas you actually need a serious one, not a trivia question. The good news is the list is finite and most of it appears in past papers again and again. The bad news is that a missing formula in your head almost always means zero method marks, because IGCSE mark schemes reward correct setup with the right relationship, not vague reasoning.

There are a few specific exceptions worth knowing. The CIE 0606 Additional Mathematics syllabus (often taken alongside 0580) does provide a printed list of formulae at the start of every paper. So if a student tells you they get a formula sheet, they are usually thinking of Additional Maths. For mainstream 0580 and for 4MA1, you are on your own.

This article gives you a pragmatic working formula sheet for Extended (the higher tier most international school students sit) with a side note on Core. It is organised by exam topic and written the way I teach it to students at International School of Milan and online: the formula, when to use it, and the trap that loses the most marks.

Algebra is the spine of every Extended paper. The formulas here are the ones you cannot derive on the spot under exam pressure.

The quadratic formula.

This is the most reused formula in the paper. If $ax^2 + bx + c = 0$ with $a \neq 0$, then

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

You need it whenever a quadratic does not factorise cleanly, which happens on most Paper 4 questions. The discriminant $b^2 - 4ac$ also tells you the number of real solutions: positive for two, zero for one, negative for none.

Difference of two squares.

$a^2 - b^2 = (a - b)(a + b)$. Examiners hide this inside fractions, surds and bigger expressions. If you see two squared terms with a minus between them, factorise.

Laws of indices.

Memorise these as a block, because mixing two of them is the most common slip:

• $a^m \times a^n = a^{m+n}$
• $a^m \div a^n = a^{m-n}$
• $(a^m)^n = a^{mn}$
• $a^0 = 1$ for $a \neq 0$
• $a^{-n} = \frac{1}{a^n}$
• $a^{1/n} = \sqrt[n]{a}$

Arithmetic and geometric sequences.

Extended expects you to find the $n$-th term of common sequences. For an arithmetic sequence with first term $a$ and common difference $d$: $u_n = a + (n - 1)d$. For a geometric sequence with first term $a$ and common ratio $r$: $u_n = a r^{n-1}$.

Compound interest.

A surprising number of marks live in financial maths. If $P$ is the principal, $r$ the rate as a percentage and $n$ the number of years:

$$A = P \left(1 + \frac{r}{100}\right)^n$$

For more on which topics carry the most marks, see how to revise for IGCSE Maths.

Mensuration questions reward the student who has the volume and surface area formulas at their fingertips. Cambridge will not give them to you.

Areas.

• Triangle: $A = \frac{1}{2} b h$ or $A = \frac{1}{2} ab\sin C$ when you have two sides and the included angle
• Parallelogram: $A = bh$
• Trapezium: $A = \frac{1}{2}(a + b)h$
• Circle: $A = \pi r^2$, circumference $C = 2\pi r$

Volumes and surface areas.

• Cuboid: $V = lwh$
• Cylinder: $V = \pi r^2 h$, curved surface area $= 2\pi r h$
• Sphere: $V = \frac{4}{3}\pi r^3$, surface area $= 4\pi r^2$
• Cone: $V = \frac{1}{3}\pi r^2 h$, curved surface area $= \pi r l$ where $l$ is the slant height
• Pyramid: $V = \frac{1}{3} \times \text{base area} \times h$

Pythagoras.

$a^2 + b^2 = c^2$ for the right triangle with hypotenuse $c$. Trivial to state, but watch for the three-dimensional version that asks for the diagonal of a cuboid: $d = \sqrt{l^2 + w^2 + h^2}$.

Trigonometry in right triangles.

SOH CAH TOA gives $\sin\theta = \frac{\text{opp}}{\text{hyp}}$, $\cos\theta = \frac{\text{adj}}{\text{hyp}}$, $\tan\theta = \frac{\text{opp}}{\text{adj}}$.

Non-right triangles (Extended only).

The sine rule and cosine rule appear on Paper 4 almost every session:

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

$$c^2 = a^2 + b^2 - 2ab\cos C$$

Arc length and sector area.

For a sector with angle $\theta$ in degrees:

$$\text{arc} = \frac{\theta}{360} \times 2\pi r, \quad \text{sector area} = \frac{\theta}{360} \times \pi r^2$$

A trap worth flagging: if the question gives the angle in radians (rare in IGCSE but it happens in Additional Maths) the formulas simplify to $r\theta$ and $\frac{1}{2}r^2\theta$. Read the question.

Statistics is where students lose marks for forgetting one bracket or one division. Keep these tight.

Mean of grouped data.

When the data is in classes, use the midpoint $x_i$ of each class:

$$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$

Probability of independent events.

$P(A \text{ and } B) = P(A) \times P(B)$. For mutually exclusive events, $P(A \text{ or } B) = P(A) + P(B)$.

Tree diagrams.

Not a formula, but a method examiners reward heavily. Multiply along the branches, add between branches for the or outcomes.

Distance, speed and time.

$\text{speed} = \frac{\text{distance}}{\text{time}}$. Memorise the triangle if it helps, but in Paper 4 you will see it dressed up inside a travel graph with several stages.

Gradients of straight lines.

Given $(x_1, y_1)$ and $(x_2, y_2)$:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

The equation of a line in the form $y = mx + c$ with gradient $m$ and intercept $c$ is the workhorse of coordinate geometry. Perpendicular lines satisfy $m_1 m_2 = -1$.

Vectors.

Magnitude of a column vector $\begin{pmatrix} a \\ b \end{pmatrix}$ is $\sqrt{a^2 + b^2}$. Parallel vectors are scalar multiples of each other.

If you want a tour of how these appear on different papers, IGCSE Maths Paper 2 vs Paper 4 breaks down where each topic typically sits.

A list of thirty formulas is not a study plan. The problem is not having seen them, it is being able to retrieve the right one under pressure when the question is dressed up.

Group by triggers, not by topic.

Instead of memorising that the circle area is $\pi r^2$, memorise that if the question gives a radius and asks about flat space, you reach for $\pi r^2$. Triggers make retrieval automatic.

Active recall, not rereading.

Cover the right side of the page and write the formula from the topic name. If you cannot, you have not learned it. Rereading creates the illusion of fluency.

Spaced practice over a month.

Three sessions a week of 15 minutes each, doing one mixed quiz across topics, beats one hour the night before the exam. The forgetting curve is real.

Past papers under timed conditions.

This is where the formulas stop being a list and start being a tool. You will quickly notice that the same five or six formulas account for most marks in Paper 4. For a structured approach see the IGCSE Maths preparation guide.

Build your own one-page sheet.

Not to take into the exam, but to revise from. The act of writing it forces you to organise the formulas the way you actually use them. Stick it on the wall above your desk and rewrite it from memory once a week.

Here is a typical Extended Paper 4 question, simplified slightly. It needs three formulas chained together.

*A cone has a base radius of 5 cm and a slant height of 13 cm. Find (a) the perpendicular height, (b) the volume, (c) the total surface area including the base.*

Step 1: perpendicular height.

The slant height $l$, radius $r$ and height $h$ form a right triangle, so by Pythagoras $l^2 = r^2 + h^2$, giving $h = \sqrt{l^2 - r^2} = \sqrt{169 - 25} = \sqrt{144} = 12$ cm.

Step 2: volume.

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

$$V = \frac{1}{3} \times \pi \times 25 \times 12 = 100\pi \approx 314.16 \text{ cm}^3$$

Step 3: total surface area.

Curved surface is $\pi r l = \pi \times 5 \times 13 = 65\pi$. The circular base is $\pi r^2 = 25\pi$. Total is $90\pi \approx 282.74$ cm$^2$.

Three different formulas, one question, full marks. None of them were on a sheet in front of you. That is what IGCSE Maths is asking for.