GCSE Higher — Geometry and measures
20% of a GCSE Higher entry is geometry and measures, spread across all three papers rather than gathered into one. The DfE lists 25 content statements for geometry and measures at this tier, and the question bank is organised by them rather than by chapter headings, so you can see exactly which one you are weak on. Expect geometric terms, notation and diagrams, ruler and compass constructions and loci, angle facts, parallel lines and polygons, properties of triangles and quadrilaterals, congruence criteria for triangles and geometric reasoning and simple proofs. Higher only: combinations of transformations and invariance, circle theorems, the sine rule and the cosine rule and area of a triangle: ½ab sin C. A Foundation entry is never asked for these. 40% of the marks are AO1, for applying a standard technique correctly, and the remaining 60% are for reasoning and for solving problems — so an answer with no working rarely scores everything that was available. Most of this area is natural Paper 1 material, so practise it without a calculator before you practise it with one.
- G1 — Geometric terms, notation and diagrams
- G2 — Ruler and compass constructions and loci
- G3 — Angle facts, parallel lines and polygons
- G4 — Properties of triangles and quadrilaterals
- G5 — Congruence criteria for triangles
- G6 — Geometric reasoning and simple proofs
- G7 — Transformations: rotation, reflection, translation, enlargement (part Higher)
- G8 — Combinations of transformations and invariance (Higher only)
- G9 — Circle definitions and parts of a circle
- G10 — Circle theorems (Higher only)
- …and 15 more geometry and measures statements at this tier
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Geometry and measures at GCSE Higher: the 25 DfE statements
Every question on this site is filed against one of these statements. Pick one to practise it on its own.
Sample geometry and measures questions for GCSE Higher
- A water tank is a cube with edges of length 2 m. Work out how many cubic centimetres the tank holds when it is full.(a)8,000,000 cm³(b)8,000 cm³(c)8 cm³(d)80,000 cm³
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8,000,000 cm³ — Method: change the edge length into centimetres first and then cube it, because 1 m = 100 cm and a volume needs that conversion applied to all three dimensions. Working: 2 m = 2 × 100 = 200 cm, so the volume is 200 × 200 × 200. 200 × 200 = 40,000 and 40,000 × 200 = 8,000,000. Answer: 8,000,000 cm³. The distractors: 8,000 cm³ comes from converting 2 m to 20 cm and cubing that; 80,000 cm³ comes from cubing in metres to get 8 m³ and then multiplying by 10,000, the conversion factor for an area rather than the 1,000,000 a volume needs; 8 cm³ comes from cubing the 2 without converting at all and simply writing cm³ because the question asked for that unit. - A wooden cube has edges of length 4 cm. Work out the total surface area of the cube.(a)96 cm²(b)24 cm²(c)16 cm²(d)64 cm³
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96 cm² — Method: a cube has six identical square faces, so the total surface area is six times the area of one face. Working: one face has area 4 × 4 = 16 cm², and 6 × 16 = 96. Answer: 96 cm². The distractors: 16 cm² is the area of a single face, from stopping before multiplying by the six faces; 64 cm³ comes from working out the volume, 4 × 4 × 4, which is a different measure and carries a different unit; 24 cm² comes from multiplying the six faces by the edge length, 6 × 4, instead of by the area of a face. - In triangle ABC, angle A is 2x°, angle B is 3x° and angle C is 4x°. Work out the size of angle B.(a)60°(b)20°(c)120°(d)80°
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60° — Method: the angles of a triangle add up to 180°, so add the three expressions, solve for x and then substitute back into the expression for angle B. Working: 2x + 3x + 4x = 9x, so 9x = 180 and x = 20. Angle B is 3x, so angle B = 3 × 20 = 60. Answer: 60°. The distractors: 20° is the value of x, from stopping as soon as the equation is solved instead of substituting back; 120° comes from using 360° as the angle sum, which gives x = 40 and 3x = 120; 80° is 4x, the angle at C, from substituting into the wrong expression. - A regular polygon has 12 sides. Work out the size of one exterior angle of the polygon.(a)30°(b)150°(c)15°(d)36°
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30° — Method: the exterior angles of any convex polygon add up to 360°, and in a regular polygon they are all equal, so divide 360° by the number of sides. Working: 360 ÷ 12 = 30. Answer: 30°. The distractors: 150° is the interior angle, 180 − 30, which answers for the wrong angle at the vertex; 15° comes from dividing 180 by 12, using the angles on a straight line instead of the full turn; 36° comes from dividing 360 by 12 − 2 = 10, carrying the subtraction of 2 out of the interior angle sum formula into a calculation that does not need it. - In triangle ABC the angle at A and the angle at C are equal. The side AB is extended beyond B, and the exterior angle formed at B measures 98°. Work out the size of the angle at A.(a)49°(b)41°(c)82°(d)98°
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49° — Method: an exterior angle of a triangle equals the sum of the two interior angles that are not next to it, which here are the angles at A and at C; since those two are equal, the exterior angle is twice the angle at A. Working: 2 × angle A = 98, so angle A = 98 ÷ 2 = 49. Answer: 49°. The distractors: 82° is the interior angle at B, 180 − 98, given in place of the angle at A; 41° comes from finding that interior angle of 82° and halving it, 82 ÷ 2, instead of halving the exterior angle; 98° comes from taking the exterior angle to be equal to the angle at A on its own, with no halving at all.
Frequently asked questions
How much of the GCSE Higher paper is geometry and measures?
20% of the total marks, and the three papers each carry an equal share of the qualification, so it is spread across all of them rather than concentrated in one.
How many topics are there in geometry and measures at GCSE Higher?
25 DfE content statements, coded G1 to G25 within this area. Every question in the bank is tagged to one of them.
Which geometry and measures topics are Higher only?
Combinations of transformations and invariance (G8), Circle theorems (G10), The sine rule and the cosine rule (G22) and Area of a triangle: ½ab sin C (G23). A further 4 statements have a Higher-only part inside otherwise shared content.
Can I use a calculator for geometry and measures questions?
Paper 1 is non-calculator; Papers 2 and 3 allow one. Of the 25 statements in this area, 16 are naturally non-calculator and 5 naturally calculator, with the rest workable either way — so practise both.
What kind of marks does geometry and measures carry?
The GCSE Higher split is 40% AO1 (use and apply standard techniques), 30% AO2 (reason, interpret and communicate) and 30% AO3 (solve problems in and out of context). Questions in this area are set across all three.
Are these past paper questions?
No. Every question at /gcse-higher is original, written to the DfE content statements and checked before it is published. We do not host past papers or mark schemes.
Do I need an account?
No — you can start practising straight away. Progress is saved automatically in your browser.
Is it free?
Yes. The questions, worked answers and printable worksheets are all free, with no adverts.
Tips for geometry and measures at GCSE Higher
- Know the circle theorems by name and be ready to quote them: the angle at the centre is twice the angle at the circumference; the angle in a semicircle is 90°; angles in the same segment are equal; opposite angles of a cyclic quadrilateral add to 180°; a tangent meets a radius at 90°; the alternate segment theorem.
- Use the sine rule a ÷ sin A = b ÷ sin B when you know a side and its opposite angle. Use the cosine rule a² = b² + c² − 2bc cos A when you know two sides and the included angle (SAS) or all three sides (SSS).
- Area of any triangle = ½ab sin C, where C is the angle between sides a and b. It replaces ½ × base × height when the height is not given.
- In coordinate geometry the three formulae are distance √((x₂ − x₁)² + (y₂ − y₁)²), midpoint ((x₁ + x₂)/2, (y₁ + y₂)/2) and gradient (y₂ − y₁) ÷ (x₂ − x₁). Parallel lines have equal gradients; perpendicular lines have gradients that multiply to −1.
- A circle with centre the origin and radius r has equation x² + y² = r². The tangent at a point is perpendicular to the radius to that point — use that to find its gradient.
Common mistakes (and how to avoid them)
- Applying the sine rule to a triangle where the given information is SAS — the cosine rule is needed there.
- Forgetting the −2bc cos A term, or getting its sign wrong, and turning the cosine rule into Pythagoras.
- Writing m₁ = m₂ for perpendicular lines instead of m₁ × m₂ = −1 (or the other way round).
- Quoting a circle theorem without naming it — a correct answer with no reason loses the reasoning marks.
Worked examples
- Two sides and the included angle are known (SAS), so use the cosine rule: BC² = AB² + AC² − 2 × AB × AC × cos A.
- BC² = 7² + 5² − 2 × 7 × 5 × cos 60° = 49 + 25 − 70 × 0.5.
- BC² = 74 − 35 = 39, so BC = √39 ≈ 6.24 cm.
- The given line has gradient 2, so the perpendicular gradient m satisfies 2m = −1, giving m = −½.
- Use y − y₁ = m(x − x₁) with (2, 3): y − 3 = −½(x − 2).
- Expand: y − 3 = −½x + 1, so y = −½x + 4.
- The angle at the centre is twice the angle at the circumference standing on the same arc.
- Angle ABC = 140° ÷ 2 = 70°.
Geometry and measures is a fifth of the Higher paper, and its Higher-only topics — circle theorems, the sine and cosine rules, ½ab sin C, coordinate geometry of circles — each have their own habit of thought. Circle theorem questions are won by naming the theorem you use; trigonometry questions are won by choosing the right rule for the information given. Start every question with a labelled sketch: it shows you which formula fits, and it stops you from putting a side or an angle in the wrong place.
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