GCSE Maths — All Topics, Formulas & Revision Guide

GCSE Maths covers number, algebra, ratio and proportion, geometry and measures, probability, and statistics. If you want a clear revision guide, focus first on the topics that appear across many papers: fractions and percentages, algebra basics, graphs, geometry formulas, and data interpretation.

The exact content can vary by exam board and tier, so treat this page as a practical map of the course and then check your own specification for the final detail.

GCSE Maths Topics At A Glance

Number

This includes place value, negative numbers, fractions, decimals, percentages, powers, roots, standard form, bounds, and estimation.

These skills show up almost everywhere else. If fractions or percentage multipliers feel shaky, harder questions in other topics often become harder than they need to be.

Algebra

Algebra usually includes simplifying expressions, substitution, expanding and factorising, solving equations and inequalities, sequences, graphs, simultaneous equations, and quadratics.

The key idea is structure. Algebra lets you describe patterns and relationships in a form you can solve.

Ratio, Proportion, And Rates of Change

This area includes ratio sharing, direct and inverse proportion, unit conversions, speed, density, and similar ideas involving rates.

A useful check is to ask what changes together. If two quantities grow in a linked way, proportion is often involved.

Geometry And Measures

This includes angle facts, perimeter, area, volume, surface area, transformations, constructions, loci, scale drawings, circles, Pythagoras, and trigonometry.

Many geometry errors come from using the right formula on the wrong shape, or using a formula correctly but in the wrong units.

Probability And Statistics

This includes probability scales, relative frequency, tree diagrams, averages, charts, cumulative frequency, box plots, histograms, and scatter graphs.

This part is about interpreting information as well as calculating it. A correct number with the wrong conclusion still loses marks.

Which GCSE Maths Topics Matter Most For Revision

If you are building a revision checklist, start here:

• Fractions, decimals, percentages, and percentage change
• Ratio, proportion, and unit rates
• Indices, roots, and standard form
• Algebraic manipulation and substitution
• Linear equations, inequalities, and simultaneous equations
• Sequences and graphs
• Area, perimeter, surface area, and volume
• Angle rules, polygons, and circle measures
• Pythagoras and trigonometry in right-angled triangles
• Probability methods and statistical diagrams

For Higher tier students, the list often extends to quadratic graphs, algebraic fractions, surds, and harder geometry or probability. The exact boundary depends on the course you are taking.

If you want an efficient order, revise number fluency first, then algebra, then percentages and ratio, then geometry formulas, then probability and statistics. Later topics often depend on earlier ones.

Key GCSE Maths Formulas To Know

Formula sheets and what is given in the exam can vary by board or year, so do not assume every formula below will always be provided. These are still worth knowing because they come up often.

Percentages And Multipliers

Percentage multiplier:

$$\text{new amount} = \text{original amount} \times \text{multiplier}$$

For example, a $12\%$ increase means multiply by $1.12$. A $12\%$ decrease means multiply by $0.88$.

Speed And Density

Two common measure formulas are:

$$\text{speed} = \frac{\text{distance}}{\text{time}}$$ $$\text{density} = \frac{\text{mass}}{\text{volume}}$$

Rearrange them only when the quantity definitions make sense in the question.

Area And Volume

Rectangle area:

$$A = lw$$

Triangle area:

$$A = \frac{1}{2}bh$$

Trapezium area:

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

Circle area:

$$A = \pi r^2$$

Prism volume:

$$V = \text{cross-sectional area} \times \text{length}$$

Circles

Circumference:

$$C = 2\pi r = \pi d$$

Be careful not to mix circumference with area. One measures distance around a circle, and the other measures space inside it.

Pythagoras And Trigonometry

For a right-angled triangle with hypotenuse $c$:

$$a^2 + b^2 = c^2$$

In a right-angled triangle only:

$$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}, \qquad \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \qquad \tan \theta = \frac{\text{opposite}}{\text{adjacent}}$$

The condition matters. These trigonometric ratios in this form are for right-angled triangles only.

Worked Example: Using Pythagoras Correctly

A right-angled triangle has one side of length $6$ cm and another side of length $8$ cm. Find the hypotenuse.

This is a Pythagoras question because:

• the triangle is right-angled
• two sides are known
• the missing side is the longest side

Use

$$a^2 + b^2 = c^2$$

Substitute the known lengths:

$$6^2 + 8^2 = c^2$$ $$36 + 64 = c^2$$ $$100 = c^2$$ $$c = 10$$

So the hypotenuse is $10$ cm.

The main lesson is the method choice. In GCSE Maths, many marks come from recognizing which tool fits the question before you calculate.

Common GCSE Maths Mistakes

Memorizing A Formula Without Knowing The Condition

Knowing $a^2 + b^2 = c^2$ is not enough. You also need to notice that it applies only to right-angled triangles.

Mixing Up Similar Ideas

Students often mix up:

• area and perimeter
• mean and median
• probability and relative frequency
• ratio simplification and ratio sharing

These pairs sound close but answer different questions.

Losing Marks On Units

If a question changes from cm to m, or from minutes to hours, the arithmetic can be correct and the final answer can still be wrong.

Revising Only Comfortable Topics

Reading solutions to topics you already like can feel productive, but it does not move your grade much. Marks usually improve faster when you spend more time on the topics you avoid.

How To Revise GCSE Maths Efficiently

Use a simple loop:

1. Pick one topic, such as percentage change or simultaneous equations.
2. Review one method or formula and do one clean worked example.
3. Answer a short mixed set of questions without notes.
4. Mark your work and write down the exact mistake type.
5. Revisit the topic a few days later.

This works better than long unfocused sessions because GCSE Maths depends heavily on recall plus method choice under time pressure.

If you are close to the exam, prioritize weak topics that appear often: percentages, algebra, graphs, geometry formulas, and data interpretation.

Where GCSE Maths Is Used

GCSE Maths is used any time you need to work with quantity, change, shape, uncertainty, or data. In everyday life that can mean budgeting, comparing deals, reading graphs, measuring spaces, or interpreting risk. In school and work, it supports subjects such as science, economics, business, and computing.

How much of the course you use later depends on what you study or do next, but the core habits stay useful: checking assumptions, choosing a method, and testing whether an answer is reasonable.

Try Your Own Revision Map

Try your own version of this guide by splitting a sheet of paper into the five big areas: number, algebra, ratio, geometry, and statistics. Under each one, list the topics you can do confidently, the formulas you forget, and one type of question you still miss. That turns revision from "study maths" into a short list you can actually finish.

If you want a next step after that, try solving a similar problem on your weakest topic and then compare your method with a clean worked solution.