AQA GCSE Maths: Geometry and Measures Revision Guide

Geometry and Measures accounts for approximately 15--20% of the marks on AQA GCSE Maths papers. It covers a wide range of topics -- from basic angle facts and properties of shapes through to trigonometry and vectors at the Higher tier. What makes this strand particularly rewarding to revise is that many geometry questions follow predictable patterns. If you learn the key facts, formulae, and methods, you can pick up marks efficiently across all three papers.

Geometry questions also tend to be multi-step. A single 4- or 5-mark question might require you to use angle facts, then apply Pythagoras' theorem, then calculate an area. This means there are plenty of method marks available for students who show clear, structured working -- even if the final answer contains a small error. This guide covers every major topic in the Geometry and Measures strand and explains how to approach each one in the exam.

Angles

Angle facts are the foundation of almost every geometry question. You need to know them fluently and be able to cite them by name when a question asks you to "give a reason."

Basic Angle Facts

• Angles on a straight line add up to 180 degrees.
• Angles around a point add up to 360 degrees.
• Vertically opposite angles are equal. When two straight lines cross, the angles opposite each other are the same size.

Angles in Parallel Lines

When a transversal crosses two parallel lines, three angle relationships are created:

• Alternate angles (sometimes called Z-angles) are equal. They appear on opposite sides of the transversal, between the parallel lines.
• Corresponding angles (sometimes called F-angles) are equal. They appear on the same side of the transversal, one between and one outside the parallel lines.
• Co-interior angles (sometimes called C-angles or allied angles) add up to 180 degrees. They appear on the same side of the transversal, between the parallel lines.

In the exam, always state which angle rule you are using. Writing "alternate angles are equal" or "co-interior angles sum to 180 degrees" is what earns you the reasoning marks.

Angles in Polygons

The sum of interior angles in any polygon is (n - 2) x 180 degrees, where n is the number of sides. For a triangle this gives 180 degrees, for a quadrilateral 360 degrees, for a pentagon 540 degrees, and so on.

The sum of exterior angles of any convex polygon is always 360 degrees. For a regular polygon, each exterior angle is 360 divided by the number of sides, and each interior angle is 180 minus the exterior angle.

Bearings

Bearings are measured clockwise from north and always written as three figures. A bearing of 45 degrees is written as 045 degrees. When finding the bearing from point A to point B, you must measure the angle at A, clockwise from north. Back bearings differ by 180 degrees -- if the bearing from A to B is 070 degrees, the bearing from B to A is 250 degrees.

Properties of Shapes

Triangles

• Scalene -- no equal sides, no equal angles.
• Isosceles -- two equal sides and two equal base angles.
• Equilateral -- all sides equal, all angles 60 degrees.
• Right-angled -- one angle of 90 degrees.

Quadrilaterals

You need to know the properties of each quadrilateral, including sides, angles, diagonals, and lines of symmetry:

• Square -- four equal sides, four right angles, diagonals bisect each other at right angles.
• Rectangle -- opposite sides equal, four right angles, diagonals bisect each other but not at right angles.
• Parallelogram -- opposite sides equal and parallel, opposite angles equal, diagonals bisect each other.
• Rhombus -- four equal sides, opposite angles equal, diagonals bisect each other at right angles.
• Trapezium -- one pair of parallel sides.
• Kite -- two pairs of adjacent equal sides, one pair of opposite angles equal, one diagonal bisects the other at right angles.

Congruence

Two shapes are congruent if they are exactly the same shape and size. For triangles, the conditions for congruence are:

• SSS -- three sides are equal.
• SAS -- two sides and the included angle are equal.
• ASA -- two angles and the included side are equal.
• RHS -- a right angle, the hypotenuse, and one other side are equal.

In proof questions, you must clearly state which condition you are using and identify the specific sides or angles that satisfy it.

Similarity

Two shapes are similar if they are the same shape but different sizes. All corresponding angles are equal, and corresponding sides are in the same ratio. The scale factor for length, area, and volume follows a specific pattern:

• If the linear scale factor is k, then the area scale factor is k squared.
• If the linear scale factor is k, then the volume scale factor is k cubed.

For example, if two similar shapes have a length ratio of 2:5, their area ratio is 4:25 and their volume ratio is 8:125. This relationship between linear, area, and volume scale factors is tested frequently on Higher tier papers.

Area, Perimeter, and Volume

2D Shapes

You need to know these area formulae without being given them:

• Rectangle -- length x width.
• Triangle -- half x base x perpendicular height.
• Parallelogram -- base x perpendicular height.
• Trapezium -- half x (a + b) x h, where a and b are the parallel sides and h is the perpendicular height.
• Circle -- pi x r squared, where r is the radius.

Circumference of a circle is 2 x pi x r (or pi x d, where d is the diameter).

Arc Length and Sector Area

An arc is a fraction of the circumference, and a sector is a fraction of the area. The fraction is determined by the angle at the centre:

• Arc length = (angle / 360) x 2 x pi x r.
• Sector area = (angle / 360) x pi x r squared.

These formulae also apply to finding the perimeter of a sector -- remember to add the two radii to the arc length.

3D Shapes -- Volume and Surface Area

• Prism -- volume = cross-sectional area x length. Surface area is the sum of all faces.
• Cylinder -- volume = pi x r squared x h. Surface area = 2 x pi x r x h + 2 x pi x r squared.
• Cone (Higher) -- volume = (1/3) x pi x r squared x h. Curved surface area = pi x r x l, where l is the slant height.
• Sphere (Higher) -- volume = (4/3) x pi x r cubed. Surface area = 4 x pi x r squared.
• Pyramid -- volume = (1/3) x base area x perpendicular height.
• Frustum (Higher) -- a cone with the top cut off. Find the volume of the full cone, subtract the volume of the small cone removed.

The formulae for cones and spheres are given on the AQA formula sheet, but you should practise using them so you can apply them quickly and accurately under timed conditions.

Pythagoras' Theorem

Pythagoras' theorem states that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: a squared + b squared = c squared, where c is the hypotenuse.

Finding the Hypotenuse

If the two shorter sides are 5 cm and 12 cm, then c squared = 25 + 144 = 169, so c = 13 cm.

Finding a Shorter Side

If the hypotenuse is 10 cm and one shorter side is 6 cm, then the missing side squared = 100 - 36 = 64, so the missing side = 8 cm.

Applications

Pythagoras' theorem appears in many contexts beyond simple triangles:

• Distance between two coordinates. The distance between (x1, y1) and (x2, y2) is the square root of (x2 - x1) squared + (y2 - y1) squared. This is Pythagoras applied to the horizontal and vertical differences.
• 3D Pythagoras (Higher). To find the longest diagonal of a cuboid with dimensions a, b, and c, use the square root of a squared + b squared + c squared. Apply Pythagoras twice -- first across the base, then from that result up to the opposite corner.
• Combining with other skills. You might need to use Pythagoras to find a missing length before calculating an area or applying trigonometry. Always draw and label the right-angled triangle you are working with.

Trigonometry

SOHCAHTOA -- Right-Angled Triangles

For any right-angled triangle, label the sides relative to the angle you are working with:

• Opposite -- the side opposite the angle.
• Adjacent -- the side next to the angle (not the hypotenuse).
• Hypotenuse -- the longest side, opposite the right angle.

The three trigonometric ratios are:

• sin(angle) = opposite / hypotenuse.
• cos(angle) = adjacent / hypotenuse.
• tan(angle) = opposite / adjacent.

To find a missing side, rearrange the appropriate formula. To find a missing angle, use the inverse function (sin inverse, cos inverse, or tan inverse). Always check that your calculator is in degree mode.

Exact Trigonometric Values

On Higher tier, you must know the exact values for key angles without a calculator:

• sin 30 = 1/2, cos 30 = (root 3)/2, tan 30 = 1/(root 3) or (root 3)/3.
• sin 45 = (root 2)/2, cos 45 = (root 2)/2, tan 45 = 1.
• sin 60 = (root 3)/2, cos 60 = 1/2, tan 60 = root 3.

These values appear frequently on Paper 1 where you cannot use a calculator to evaluate trigonometric functions.

Sine Rule, Cosine Rule, and Area Formula (Higher)

For non-right-angled triangles, three additional tools are available:

• Sine rule -- a / sin A = b / sin B = c / sin C. Use this when you know a side and its opposite angle plus one other piece of information.
• Cosine rule -- a squared = b squared + c squared - 2bc cos A. Use this when you know two sides and the included angle, or all three sides and need to find an angle.
• Area formula -- area = (1/2) x a x b x sin C. Use this when you know two sides and the included angle.

3D Trigonometry (Higher)

In 3D problems, identify the right-angled triangle within the 3D shape. Draw it out separately, label the sides, and apply Pythagoras or trigonometry. The key is to break the 3D problem down into 2D triangles.

Transformations

Every transformation question requires you to either perform a transformation or describe one. Descriptions must be complete -- missing any required detail loses marks.

Translation

A translation moves a shape without rotating, reflecting, or resizing it. It is described by a column vector. The top number gives the horizontal movement (positive = right, negative = left) and the bottom number gives the vertical movement (positive = up, negative = down).

To describe a translation, you must state "translation" and give the column vector. Both are required for full marks.

Reflection

A reflection creates a mirror image of a shape in a given line. To describe a reflection, you must state "reflection" and give the equation of the mirror line (for example, "reflection in the line y = x" or "reflection in the line x = -1").

Rotation

A rotation turns a shape around a fixed point. To describe a rotation, you must state "rotation," the angle of rotation, the direction (clockwise or anticlockwise), and the centre of rotation. All four details are required. If the angle is 180 degrees, you do not need to state the direction.

Enlargement

An enlargement changes the size of a shape from a fixed centre. To describe an enlargement, you must state "enlargement," the scale factor, and the centre of enlargement.

• A scale factor greater than 1 makes the shape larger.
• A scale factor between 0 and 1 makes the shape smaller.
• A negative scale factor (Higher) inverts the shape through the centre of enlargement, producing an image on the opposite side.
• A fractional scale factor (Higher) produces a smaller shape.

When performing an enlargement, measure the distance from the centre to each vertex, multiply by the scale factor, and plot the new vertices at those distances from the centre.

Vectors (Higher)

Vectors represent a quantity with both magnitude and direction. In GCSE Maths, vectors are used to describe movements and to prove geometric results.

Notation

A vector from point A to point B is written as the vector AB (with an arrow above) or as a bold lowercase letter such as a. In handwriting, underline the letter instead of using bold.

Adding and Subtracting Vectors

To travel from A to C via B, you add the vectors: vector AC = vector AB + vector BC. To reverse a vector, negate it: vector BA = negative of vector AB.

Scalar Multiples

Multiplying a vector by a scalar changes its magnitude but not its direction (unless the scalar is negative, in which case the direction reverses). If vector AB = 2a, then the line AB is twice as long as a and in the same direction.

Proving Collinearity

To prove that three points are collinear (lie on the same straight line), show that one vector between two of the points is a scalar multiple of a vector between another pair. If vector AB = k x vector AC for some scalar k, then A, B, and C are collinear.

Finding Ratios

Vectors are often used to find the ratio in which a point divides a line segment. Express the vectors from a common origin and use the relationship between them to determine the ratio.

Constructions and Loci

Constructions

You need to be able to perform the following constructions accurately using only a ruler and compasses:

• Constructing a triangle given three sides (SSS). Draw one side, then use compasses set to the lengths of the other two sides to find the third vertex.
• Perpendicular bisector of a line segment. Open your compasses to more than half the length of the segment, draw arcs from both endpoints, and connect the intersection points.
• Angle bisector. Draw an arc from the vertex to create two points on the arms of the angle, then draw arcs from those two points. The line from the vertex through the intersection of these arcs bisects the angle.
• Perpendicular from a point to a line. Use compasses to find two equidistant points on the line, then construct the perpendicular bisector of those two points.

Always leave your construction arcs visible. The examiner needs to see that you used compasses, not freehand drawing.

Loci

A locus is a set of points satisfying a given condition:

• Equidistant from a fixed point -- a circle centred on that point.
• Equidistant from a line -- a pair of parallel lines either side of the original line, with semicircles at the ends.
• Equidistant from two fixed points -- the perpendicular bisector of the line segment joining them.
• Equidistant from two intersecting lines -- the angle bisectors of the lines.

Loci questions often combine multiple conditions. The answer is the region where all conditions overlap. Shade or mark this region clearly.

Common Mistakes

Geometry questions are among the most error-prone on the paper. Here are the mistakes examiners see most frequently:

• Forgetting units. If your answer is a length in centimetres or a volume in cubic metres, write the units. Omitting them can cost you the final accuracy mark.
• Using the wrong formula. Double-check whether you need area or perimeter, circumference or area, volume or surface area. Read the question twice.
• Not labelling diagrams. When you calculate a length or angle, write it on the diagram. This helps you and the examiner follow your working.
• Confusing trigonometric rules. Know when to use SOHCAHTOA (right-angled triangles only) versus the sine rule or cosine rule (non-right-angled triangles).
• Incomplete transformation descriptions. Every transformation type has a specific set of required details. Missing any one of them -- such as omitting the centre of rotation or the mirror line equation -- will lose marks.
• Rounding too early in trigonometric calculations. Keep full decimal values on your calculator throughout the calculation and only round at the final step. Early rounding introduces errors that accumulate through multi-step problems.
• Mixing up radius and diameter. Many students accidentally use the diameter in a formula that requires the radius, or vice versa. Always check which one the question has given you.

Prepare with LearningBro

LearningBro's GCSE Mathematics Geometry course covers every topic in this guide with structured lessons and exam-style practice questions. Each question is designed to mirror what you will see on the real AQA papers, so you build familiarity with the question formats alongside your mathematical knowledge.

For broader exam preparation, our AQA GCSE Maths Exam Prep course covers all strands of the specification, including number, algebra, ratio, statistics, and geometry. You can also read our AQA GCSE Maths Exam Technique guide for strategies on showing working, managing time, and maximising method marks across all three papers.

Geometry rewards students who practise consistently and show clear working. Learn the angle facts, memorise the formulae, and always draw and label your diagrams. With structured revision and the right technique, this is a strand where you can pick up marks reliably on every paper.

Good luck with your revision.