AQA GCSE Foundation Maths Formula Sheet

Before you start

AQA provides official specification and formulae information for GCSE Mathematics. This article is designed for revision and should be used alongside school and AQA guidance.

This article is deliberately more than a list. Formulae are easy to copy but harder to use under exam pressure, so each section includes a short note about the situation where the formula normally appears.

How to use this formula sheet

This AQA Foundation formula sheet is most useful when it is used actively. Do not just read the formulae from top to bottom. Choose one section, cover the notes, and check whether you can explain what each letter means before you start calculating.

A good revision habit is to keep a rough page beside you. For each formula, write down the units, draw a small diagram if the topic is geometry, and complete one easy question before moving to a harder one. This builds confidence without rushing straight into the most difficult exam questions.

If a formula includes a square, cube, square root or fraction, pause before substituting. These are the places where small calculator errors often appear. Brackets are especially important for negative numbers, compound interest, the quadratic formula and trigonometry.

Quick revision checklist

• Can you write the formula without looking?
• Can you explain what each letter means?
• Can you identify the correct units for the answer?
• Can you rearrange the formula if the subject is not already the unknown?
• Can you apply the formula to an exam-style question with extra wording?
• Can you spot when two formulae are needed in the same question?

Formulae by topic

Formula-by-formula practice notes

Core Foundation formulae

Area of a rectangle: Use this for rectangles and for breaking compound shapes into simpler rectangles. In revision, practise this formula in three steps: first with numbers that are already in the correct units, then with a question where one value must be found from the diagram or text, and finally with a mixed exam-style question where you need to decide for yourself that this is the right method.

Area of a triangle: The base and height must meet at right angles. They do not have to be the sloping sides drawn on the diagram. In revision, practise this formula in three steps: first with numbers that are already in the correct units, then with a question where one value must be found from the diagram or text, and finally with a mixed exam-style question where you need to decide for yourself that this is the right method.

Area of a parallelogram: Use the perpendicular height, not the slanted side. In revision, practise this formula in three steps: first with numbers that are already in the correct units, then with a question where one value must be found from the diagram or text, and finally with a mixed exam-style question where you need to decide for yourself that this is the right method.

Area of a trapezium: The parallel sides are a and b. The height is the perpendicular distance between them. In revision, practise this formula in three steps: first with numbers that are already in the correct units, then with a question where one value must be found from the diagram or text, and finally with a mixed exam-style question where you need to decide for yourself that this is the right method.

Circumference of a circle: Use diameter d if it is given. Use twice the radius if only r is given. In revision, practise this formula in three steps: first with numbers that are already in the correct units, then with a question where one value must be found from the diagram or text, and finally with a mixed exam-style question where you need to decide for yourself that this is the right method.

Area of a circle: Square the radius before multiplying by pi. In revision, practise this formula in three steps: first with numbers that are already in the correct units, then with a question where one value must be found from the diagram or text, and finally with a mixed exam-style question where you need to decide for yourself that this is the right method.

Volume of a cuboid: Check that all measurements are in the same units before multiplying. In revision, practise this formula in three steps: first with numbers that are already in the correct units, then with a question where one value must be found from the diagram or text, and finally with a mixed exam-style question where you need to decide for yourself that this is the right method.

Volume of a prism: Find the repeated cross-section first, then multiply by the length of the prism. In revision, practise this formula in three steps: first with numbers that are already in the correct units, then with a question where one value must be found from the diagram or text, and finally with a mixed exam-style question where you need to decide for yourself that this is the right method.

Speed, distance and time: Rearrange to distance = speed x time and time = distance divided by speed. In revision, practise this formula in three steps: first with numbers that are already in the correct units, then with a question where one value must be found from the diagram or text, and finally with a mixed exam-style question where you need to decide for yourself that this is the right method.

Density, mass and volume: A unit check helps: kg/m3, g/cm3 and similar units tell you which quantities are being divided. In revision, practise this formula in three steps: first with numbers that are already in the correct units, then with a question where one value must be found from the diagram or text, and finally with a mixed exam-style question where you need to decide for yourself that this is the right method.

Mean: For grouped data, use midpoints to estimate the total. In revision, practise this formula in three steps: first with numbers that are already in the correct units, then with a question where one value must be found from the diagram or text, and finally with a mixed exam-style question where you need to decide for yourself that this is the right method.

Range: The range is a measure of spread, not an average. In revision, practise this formula in three steps: first with numbers that are already in the correct units, then with a question where one value must be found from the diagram or text, and finally with a mixed exam-style question where you need to decide for yourself that this is the right method.

Mini practice prompts

Use these prompts to turn the formula sheet into active revision. They are not full past-paper questions, but they give you a quick way to check whether you understand the method before moving on to exam papers.

• Choose one formula and write a question where the answer is a length.
• Choose one formula and write a question where the answer is an area or volume.
• Choose one formula and rearrange it so a different letter is the subject.
• Choose one geometry formula and explain which measurement must be perpendicular.
• Choose one calculator formula and type it into your calculator using brackets.
• Choose one formula and write down the units you expect in the answer.
• Choose one formula and list two mistakes a student might make with it.
• Choose one formula and find a past-paper question that uses the same idea.

If a prompt feels too easy, make it harder by adding a unit conversion, a diagram, a decimal answer that needs rounding, or a second step before the formula can be used. That is how straightforward formula practice becomes proper exam preparation.

Using this page with past papers

Past papers are most useful when you use them to test decisions, not just answers. Before checking the mark scheme, write down which formula you chose and why. If the mark scheme uses a different method, compare the trigger in the question with the trigger you wrote down. This helps you become faster at recognising methods next time.

For each mistake, keep a short correction log. A useful correction is specific: 'I used diameter instead of radius', 'I rounded the answer before the final step', or 'I chose cosine when the opposite and hypotenuse were involved'. A vague correction like 'revise circles' is harder to act on.

Once a week, return to the same formulae in a different order. Exams mix topics, so revision should mix topics too. A student who can use a formula only when the worksheet title gives the topic away is not yet exam-ready with that formula.

Parents can help by asking students to explain the method aloud. The explanation does not need to be perfect mathematical language. It just needs to show that the student knows what the formula does, which numbers go where, and what the final answer means.

It is also worth alternating calculator and non-calculator practice where the topic allows it. Calculator questions test accurate substitution, brackets and rounding. Non-calculator questions test structure, simplification and number sense. Strong students can usually move between both without treating them as completely separate skills.

If you are short on time, do not try to revise every formula with equal weight in one sitting. Pick the formulae that appear most often in your recent classwork or mock-paper mistakes, practise those properly, and then widen the list once the weakest areas feel steadier.

The final check is confidence under time pressure. Set a short timer, attempt two mixed questions, and then review the method calmly and honestly.

How to turn formula knowledge into marks

Formulae only become useful when you can recognise the situation in the question. Exam boards rarely ask, 'Use this formula now.' Instead, they describe a shape, graph, sequence, financial situation or data set, and you have to decide which relationship connects the information.

Start by underlining the quantities you are given. Then write the formula before substituting numbers. This gives the examiner a clear method and gives you a chance to notice whether the units or variables make sense.

For geometry, draw on the diagram. Mark the radius, diameter, perpendicular height, slant height, hypotenuse or included angle. Many students know the formula but use the wrong length because the diagram has not been labelled carefully.

For algebra, keep each line tidy. If you are rearranging, do the same operation to both sides and avoid trying to do several steps at once. Method marks often come from showing a logical process, even if the final answer goes wrong.

For calculator papers, type expressions exactly as they are written, using brackets where needed. For example, the denominator in the quadratic formula is all of 2a, and compound interest needs the growth multiplier inside brackets before applying the power.

For non-calculator papers, look for structure. Fractions may simplify, surds may need rationalising, and circle or triangle questions may be designed to use exact values rather than decimal approximations.

How to practise without just memorising

A formula sheet is a starting point, not the whole revision task. The aim is to move from recognition to retrieval to application. Recognition means the formula looks familiar. Retrieval means you can write it down without looking. Application means you can use it correctly when the question is worded differently from the example in your notes.

Many students stop at recognition because it feels comfortable. They read through the page, understand the lines, and feel that the topic is done. In an exam, though, the question may hide the formula inside a context about money, speed, a garden design, a graph, a cone, or a data set. That is why mixed practice is so important.

When you revise, put a tick beside formulae you can use independently, a question mark beside formulae you understand but sometimes forget, and a star beside formulae that regularly lead to mistakes. Your next revision session should begin with the question marks and stars, not with the formulae that already feel easy.

It can also help to write a short sentence beside each formula. For example, 'use this when I know the radius and need the area', or 'use this when I have two sides and the included angle'. Those plain-English triggers are often more useful than rewriting the algebra again and again.

If you are revising with a parent or tutor, ask them to test the trigger, not just the formula. They might ask, 'Which formula would you use for the distance around a circle?' or 'Which triangle formula works when you have two sides and the angle between them?' That style of questioning is much closer to exam thinking.

Common mistakes to avoid

• Using diameter instead of radius in circle, cone, cylinder or sphere formulae.
• Forgetting to square or cube a value before multiplying.
• Using the slant height when the formula needs the perpendicular height.
• Rounding too early and carrying an inaccurate value through the rest of the question.
• Not converting units before substituting into a formula.
• Choosing sine, cosine or tangent before labelling opposite, adjacent and hypotenuse.
• Using a formula correctly but giving the answer without units.

A simple weekly revision routine

A formula sheet should sit inside a wider revision routine. Spend one short session learning and recalling the formulae, one session applying them to mixed questions, and one session reviewing mistakes. That pattern is more effective than re-reading the page the night before a test.

If you are working towards a AQA Foundation paper, keep a separate list of formulae that feel uncertain. Bring that list to school, tutoring or independent revision so the work is targeted rather than vague.

You can also use this sheet alongside GCSE revision support, Maths videos and GCSE Maths tutoring if you want clearer examples and more structured practice.

What to do next

Choose three formulae from this page and find one exam-style question for each. Mark the questions carefully, write down any mistakes, and repeat the same formulae a few days later. That spaced practice is what helps the formulae stay useful in an exam.

If formulae are one of the areas holding you back, get in touch with Sophie for calm, focused online Maths support. Lessons can target the exact topics, methods and exam skills that need more confidence.

FAQs

Do I need to memorise every GCSE Maths formula?

You need to know how to use the important formulae confidently. In some exam years certain formulae may be provided, but that does not remove the need to recognise which formula fits the question.

What is the best way to revise formulae?

Use short, regular practice. Cover the formula, write it from memory, then apply it to two or three exam-style questions. The application stage matters more than copying a list.

Should I make flashcards for Maths formulae?

Flashcards can help, but they should include a tiny example or a note about when to use the formula. A formula without context is easy to forget in an exam.

Why do I know the formula but still get questions wrong?

Usually the issue is substitution, units, rearranging, rounding or choosing the wrong measurement from a diagram. Work through your mistakes and label each one clearly.

Are Foundation and Higher formulae different?

There is overlap, but Higher students need additional formulae and more flexible use of algebra, trigonometry and 3D geometry. Foundation students still need secure basics.

Can a tutor help with formulae and exam technique?

Yes. A tutor can help identify which formulae are secure, which ones are confused, and how to practise them in exam-style questions rather than learning them in isolation.