Geometry and measures practice problems GCSE Maths, Topic 4
Angles and polygons, area and volume, Pythagoras' theorem, right-angled trigonometry and the sine and cosine rules. Try each question before you look: use the hint if you are stuck, then reveal the worked solution one step at a time.
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Key equations for geometry and measures
| a2 + b2 = c2 | Pythagoras' theorem |
|---|---|
| sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj | trigonometry (SOH CAH TOA) |
| A = ½(a + b)h | trapezium |
| V = πr2h | cylinder |
| a2 = b2 + c2 − 2bc cos A | cosine rule (Higher) |
A formulae sheet is provided in the 2025, 2026 and 2027 GCSE Maths exams. Answers within 2.5% are marked correct, and you can type fractions such as 23/20. Every topic, Foundation and Higher, is in our GCSE Maths topic list.
Warm-up questions
Quick checks on the core ideas. If these feel easy, go straight to the exam-style questions.
Hint
Angles in a triangle add up to 180°.
Worked solution
- 180 − 50 − 65 = 65°.
Hint
A = ½(a + b)h.
Worked solution
- ½ × (6 + 10) × 4 = 32 cm².
Hint
A = πr2.
Worked solution
- A = π × 52 = 25π.
- A = 78.54 cm².
Exam-style questions
Questions at the level of the GCSE Maths exam. The ones marked Challenge combine more than one idea.
Hint
Use Pythagoras' theorem.
Worked solution
- c2 = 92 + 122 = 225.
- c = 15 cm.
Hint
Opposite and hypotenuse: use sine.
Worked solution
- sin 35° = opp/10.
- opp = 10 × sin 35° = 5.74 cm.
Hint
V = πr2h.
Worked solution
- V = π × 9 × 10 = 90π.
- V = 282.7 cm³.
Hint
Each exterior angle is 360° ÷ 6.
Worked solution
- Exterior angle = 60°.
- Interior angle = 180 − 60 = 120°.
Hint
Two sides and the angle between them: use the cosine rule.
Worked solution
- a2 = 72 + 92 − 2 × 7 × 9 × cos 60° = 49 + 81 − 63 = 67.
- a = √67 = 8.19 cm.
Exam question practice
One original exam-style question in the style of a GCSE paper. Answer it in full, with working, before you open the mark scheme: most marks are for method, so show every step.
Exam-style question style · 6 points · about 6 minutes
A ladder 6.5 m long leans against a vertical wall. The top of the ladder is 6 m above the ground, which is horizontal.
- (a) How far is the foot of the ladder from the wall?
- (b) Work out the angle between the ladder and the ground, to 1 decimal place.
Write your full answer on paper first, then open the scoring guide and tick each point you earned.
Scoring guide and model answer
Part (a)
d2 = 6.52 − 62 = 42.25 − 36 = 6.25, so d = 2.5 m.
Part (b)
sin θ = 6/6.5, so θ = 67.4°.
Your mark: 0 of 6
Common mistakes in geometry and measures
- Adding the squares when finding a shorter side: subtract from the hypotenuse squared.
- Calculator in radians instead of degrees.
- Using the diameter instead of the radius in circle formulas.
- Using the cosine rule when the sine rule is simpler, or the other way round.
Keep going
Revise the ideas behind these questions:
- Pythagoras' theorem guide
- GCSE trigonometry guide
- Sine and cosine rules guide
- Volume of a cylinder guide
- Area of a trapezium guide
Then try real exam questions: GCSE Maths past papers, with official scoring guidelines. · All GCSE Maths units