GCSE maths formulae: the higher formula sheet and what to learn

Do students get a formula sheet in GCSE maths?

Yes. Every GCSE maths exam now comes with a formula sheet, on both the foundation and higher tiers, and this has been confirmed until the current GCSEs come to an end. The sheets were first used in 2022. Before that, students had to remember these formulae, because the GCSE subject content said they must not be given in the exam.

Exam series	Formula sheet given?
June and November 2025	Yes
June and November 2026	Yes
June and November 2027	Yes
2028 onwards, until the current GCSEs end (resits included)	Yes

The Department for Education decided that students taking the current GCSE maths don't need to memorise the formulae, and Ofqual, the exams regulator, then made the sheets a rule for the exam boards. New GCSEs are planned to be first taught in 2029 or 2030, and the rules for those haven't been decided yet.

What the rules say:

• Each board publishes its sheets by 1 September the year before the exams, so students can get used to them.
• A clean copy comes with every exam paper. WJEC Eduqas prints it inside the paper itself, from summer 2026.
• The sheets stay the same from year to year and are the same for every exam board.
• No question can be answered just by copying something from the sheet.

What's on the GCSE maths higher formula sheet?

The GCSE maths higher formula sheet has these formulae, in four groups. The ones marked "higher only" aren't on the foundation sheet.

Perimeter, area and volume

• Area of a trapezium = ¹⁄₂(a + b)h, where a and b are the parallel sides and h is the height between them
• Volume of a prism = area of cross section × length
• Circumference of a circle = 2πr = πd, where r is the radius and d is the diameter
• Area of a circle = πr²

Pythagoras' theorem and trigonometry

• Pythagoras' theorem: a² + b² = c², in a right-angled triangle where c is the hypotenuse
• Trigonometry, in a right-angled triangle ABC where c is the hypotenuse: sin A = a over c, cos A = b over c, tan A = a over b
• Sine rule (higher only), in any triangle ABC: a over sin A = b over sin B = c over sin C
• Cosine rule (higher only): a² = b² + c² − 2bc cos A
• Area of a triangle (higher only) = ¹⁄₂ab sin C

The quadratic formula (higher only)

The solutions of ax² + bx + c = 0, where a isn't 0, are x = −b ± √(b² − 4ac) over 2a

Compound interest and probability

• Compound interest: total accrued = P(1 + r over 100)ⁿ, where P is the amount at the start, r is the interest rate for each period and n is the number of times the interest is added
• Probability: P(A or B) = P(A) + P(B) − P(A and B)
• Conditional probability (higher only): P(A and B) = P(A given B)P(B)

The GCSE foundation maths formula sheet

The GCSE foundation maths formula sheet has the same formulae without the five higher-only ones. So it has the trapezium, prism and circle formulae, Pythagoras' theorem, sin, cos and tan, compound interest and P(A or B). It doesn't have the sine rule, the cosine rule, the area of a triangle using sin C, the quadratic formula or the conditional probability formula, because the GCSE content that uses them is assessed on the higher tier only.

Formulae given in the question instead

A few formulae are given in the question that needs them, not on the sheet. The GCSE subject content says students should be able to use them but don't need to memorise them:

• Cones: curved surface area = πrl, where l is the slant height, and volume = ¹⁄₃πr²h
• Spheres: surface area = 4πr², and volume = ⁴⁄₃πr³
• Constant acceleration: v = u + at, s = ut + ¹⁄₂at² and v² = u² + 2as

Which GCSE maths formulae do students still need to learn?

Students still need to know the formulae that aren't on the sheet or in the question. The GCSE content expects students to know these and use them:

• Area: a rectangle is length × width, a triangle is ¹⁄₂ × base × height, and a parallelogram is base × height
• Volume: a cuboid is length × width × height. A cylinder is a prism, so its volume is πr² × length.
• Parts of a circle: arc lengths and the areas of sectors, worked out as a fraction of the whole circle
• Exact trig values: sin and cos of 0°, 30°, 45°, 60° and 90°, and tan of 0°, 30°, 45° and 60°. For example, cos 60° = 0.5.
• Compound units: speed, density and pressure, such as speed = distance ÷ time

Having the sheet doesn't take away the main skill. Students have to choose the right formula, put the numbers in correctly, and often rearrange it first. The sheet just means they don't have to remember the formula itself.

Worked examples

Foundation: the area of a trapezium

A trapezium has parallel sides of 5 cm and 9 cm, and a height of 4 cm. Using ¹⁄₂(a + b)h: first 5 + 9 = 14, then 14 × 4 = 56, and half of 56 is 28. The area is 28 cm².

Higher: the cosine rule

A triangle has sides b = 5 cm and c = 8 cm, with an angle of 60° between them. Find the third side, a. Put the numbers into a² = b² + c² − 2bc cos A:

1. 5² + 8² = 89
2. 2 × 5 × 8 = 80, and cos 60° is 0.5, so 80 × 0.5 = 40
3. 89 − 40 = 49, so a² is 49
4. √49 = 7, so a is 7 cm

Both tiers: compound interest

£2,000 is saved at 3% interest a year, compounded every year, for 2 years. Here P is 2,000, r is 3 and n is 2. 1 + 3 ÷ 100 gives 1.03, and 1.03² = 1.0609. Then 2,000 × 1.0609 = 2,121.8, so the total is £2,121.80.

Common mistakes with GCSE maths formulae

• Thinking the sheet does the work. It gives the formula, but not which one to use or how. Questions are written so that copying from the sheet isn't enough.
• Mixing up the sides in sin, cos and tan. The sheet uses a, b and c, but exam triangles use all sorts of letters. Label the opposite, adjacent and hypotenuse from the angle first.
• Forgetting the square root. Pythagoras' theorem and the cosine rule give a² or c². The length is the square root of that.
• Doing the cosine rule in the wrong order. Work out 2bc cos A first, then take it away from b² + c².
• Losing the minus sign in the quadratic formula. If b is negative, −b is positive. The whole of −b ± √(b² − 4ac) goes over 2a, not just the square root.
• A calculator in the wrong angle mode. Trigonometry answers in GCSE maths are in degrees, so check the calculator shows D or DEG.

How to learn GCSE maths formulae and use them

• Practise with the real sheet. Download it from your child's exam board, such as the AQA assessment resources or OCR assessment page, and keep it beside them for practice questions, so they know where everything is.
• Know what every letter means. For each formula, your child should be able to say what r, h, n or l stands for, without looking.
• Learn the ones that aren't on the sheet. Make a short card of the area, volume, exact trig values and compound units, and test from memory a few times a week.
• Practise substituting and rearranging. Put numbers into a formula, then make another letter the subject. Solving equations uses the same steps.
• Mix the topics. Exam questions don't say which formula they need. A mixed set of questions is practice at choosing.

The quadratic formula has its own guide in solving quadratic equations, and compound interest builds on percentage change. For how the exams are graded, see GCSE maths grade boundaries.

Questions to try

1. A trapezium has parallel sides of 6 cm and 10 cm and a height of 5 cm. What's its area?
2. A circle has a radius of 3 cm. Give its area in terms of π.
3. A right-angled triangle has shorter sides of 6 cm and 8 cm. How long is the hypotenuse?
4. Is the quadratic formula on the foundation formula sheet?
5. £500 is saved at 2% compound interest a year for 3 years. What's the total at the end?
6. A triangle has sides of 3 cm and 5 cm with an angle of 60° between them. Use the cosine rule to find the third side, to 2 decimal places.

Answers

1. 40 cm²: 6 + 10 = 16, 16 × 5 = 80, and half of 80 is 40
2. 9π cm², because 3² = 9
3. 10 cm: 6² + 8² = 100, and √100 = 10
4. No. It's on the higher sheet only, because solving quadratics with the formula is higher tier content.
5. £530.60: 1.02³ = 1.061208, and 500 × 1.061208 = 530.604
6. 4.36 cm: 3² + 5² = 34, 2 × 3 × 5 = 30, 30 × 0.5 = 15, 34 − 15 = 19, and √19 is about 4.36