GCSE Maths Formula Sheet: What You Need to Know
One of the most common questions I hear from students in the run-up to the exam is: "Which formulas will I be given, and which ones do I need to learn?" It's a good question — and the answer might surprise you, because the formula sheet is shorter than most students expect, and some formulas that come up regularly in the exam aren't on it at all.
This guide tells you exactly what you get, what you don't, and how to make sure the ones you need to memorise actually stick.
What's on the Formula Sheet
A formula sheet is provided at the start of each paper. Here's what's included on both Foundation and Higher tiers:
Area and volume:
Area of a trapezium:
$$A = \frac{1}{2}(a + b)h$$
Volume of a prism:
$$V = A_{\text{cross-section}} \times l$$
Circumference of a circle:
$$C = 2\pi r = \pi d$$
Area of a circle:
$$A = \pi r^2$$
Pythagoras' theorem:
$$a^2 + b^2 = c^2$$
Trigonometry (right-angled triangles):
$$\sin A = \frac{a}{c}, \quad \cos A = \frac{b}{c}, \quad \tan A = \frac{a}{b}$$
Compound interest:
$$\text{Total accrued} = P!\left(1 + \frac{r}{100}\right)^n$$
Probability:
$$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$$
The Higher tier sheet includes additional formulas not found on Foundation:
Higher only:
Quadratic formula:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
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 a triangle:
$$A = \frac{1}{2}ab\sin C$$
Conditional probability:
$$P(A \text{ and } B) = P(A \mid B) \cdot P(B)$$
What's NOT on the Formula Sheet
This is where a lot of students get caught out. The following formulas appear regularly in the exam but are not provided — you'll need to know them from memory.
Basic geometry:
- Area of a rectangle: $l \times w$
- Area of a triangle: $\frac{1}{2} \times \text{base} \times \text{height}$
Volume of a cone:
$$V = \frac{1}{3}\pi r^2 h$$
Volume of a sphere:
$$V = \frac{4}{3}\pi r^3$$
Surface area of a sphere:
$$SA = 4\pi r^2$$
Physics-style formulas (but tested in maths too):
- Density = mass ÷ volume
- Speed = distance ÷ time
- Pressure = force ÷ area
Algebra:
Gradient of a line:
$$m = \frac{y_2 - y_1}{x_2 - x_1}$$
Equation of a straight line: $y = mx + c$
Statistics and number:
- Mean: total ÷ number of values
- Percentage change: (change ÷ original) × 100
The density, speed, and pressure formulas catch a lot of students off guard — they feel more like science than maths, so it's easy to assume they'll be provided. They won't be. If you haven't practised them recently, try a few questions now to make sure they're solid.
How to Actually Learn These Formulas
The least effective way to learn a formula is to copy it out repeatedly without ever using it. That creates a feeling of familiarity without actually building the memory.
What works much better is using the formula to answer questions, then checking whether you got it right without looking it up. If you can work through ten questions on a topic using a formula correctly from memory, you know it. If you're still reaching for your notes after five questions, you need a bit more practice before it'll consolidate.
Bow Tie Maths lets you practise each of these topics individually with unlimited exam-style questions — so instead of copying formulas into a notebook, you can test yourself on them until they actually stick.
A few other approaches worth trying:
Explain it to someone else. This sounds simple, but having to put a formula into words — even explaining it to a family member who doesn't do maths — forces you to understand it rather than just recognise it. If you can't explain why a formula works, even loosely, you're more likely to misremember it under pressure.
Write them out before each session. Spend two minutes at the start of each practice session writing out all the formulas you need from memory before you look at any questions. It's a low-effort habit that reinforces them over time without needing to dedicate a whole revision session to it.
Connect them to where they come from. The area of a circle is $\pi r^2$ because of the relationship between radius and circumference. You don't need to know the proof, but knowing there's a reason behind the formula — rather than treating it as a random string of symbols — makes it much easier to hold onto.
Foundation vs Higher
The Higher formula sheet includes everything Foundation does, plus the quadratic formula, the sine and cosine rules, the $\frac{1}{2}ab\sin C$ area formula, and the conditional probability rule $P(A \text{ and } B) = P(A \mid B) \cdot P(B)$.
If you're sitting Foundation, you won't see those on your sheet. Everything else — Pythagoras, circle formulas, compound interest, and the basic probability rule — is provided on both tiers, so you don't need to memorise those.
If you're not certain which tier you're sitting, ask your teacher. It matters, because what you need to learn differs between the two.
The Ones Worth Double-Checking
A few formulas tend to trip students up because they're easy to mix up with something similar:
Circumference vs area of a circle. Circumference is $2\pi r$; area is $\pi r^2$. A simple way to remember: area involves $r^2$, because you're covering a two-dimensional space, while circumference is a length — just $r$ once.
Volume vs surface area of a sphere. Volume is $\frac{4}{3}\pi r^3$; surface area is $4\pi r^2$. Neither is on the formula sheet, so both need to be memorised. The exam will tell you which one it's asking for, but it's worth double-checking before you start the calculation.
Mean vs median. Mean is calculated (total ÷ count); median is the middle value once the data is in order. Students sometimes swap these under pressure, especially in a statistics question that includes both.
Density, speed, and pressure. All three follow the same structure — a quantity equals one thing divided by another — which makes them easy to mix up. Density is mass ÷ volume, speed is distance ÷ time, pressure is force ÷ area. A triangle diagram (the kind where you cover up the value you want to find) works well for all three if you're struggling to keep them straight.
Want to put all of this into practice? The student section has unlimited exam-style questions across every topic on this page — so you can check whether you've genuinely learned each formula or whether you still need a bit more work on it.