IGCSE Mathematics: Mensuration — Practice Questions

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs — examination boards hold copyright in their own papers. Use these alongside the official past papers available free from your board.

Related: Mensuration revision notes

Tier note: questions 1 to 10 are Core (0580 C5). Questions 11 to 13, marked (Extended), are Extended only (E5.3 to E5.5): Core covers sectors only where the angle is a factor of 360°, and not frustums or other parts of solids. For 11 to 13, use the π key on your calculator.

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Questions

1. Convert 3.4 m² into cm². [2]

2. A rectangle has length 9 cm and width 6 cm. Calculate its area and perimeter. [2]

3. A circle has radius 8 cm. Taking π = 3.142, calculate:

(a) the circumference of the circle. [2] (b) the area of the circle. [2]

4. A sector of a circle has radius 8 cm and a sector angle of 60°. Taking π = 3.142, calculate:

(a) the arc length of the sector. [2] (b) the area of the sector. [2]

5. A cylinder has radius 4 cm and height 10 cm. Taking π = 3.142, calculate:

(a) its volume. [2] (b) its total surface area. [3]

6. A cone has radius 3 cm and slant height 5 cm. Taking π = 3.142, calculate:

(a) the perpendicular height of the cone. [2] (b) the volume of the cone. [2]

7. A shape is made from a 10 cm × 6 cm rectangle with a semicircle of diameter 6 cm removed from one short side. Taking π = 3.142, calculate the area of the remaining compound shape. [3]

8. A sphere has radius 6 cm. Taking π = 3.142, calculate its volume, correct to 3 significant figures. [2]

9. A sphere has surface area 314.2 cm². Taking π = 3.142, find its radius. [3]

10. Calculate the area of a circle of radius 6 cm. Give your answer correct to 3 significant figures. [2]

11. (Extended) A major sector of a circle has radius 7 cm and sector angle 250°. Calculate:

(a) the arc length [2] (b) the area of the sector [2] (c) the perimeter of the sector. [1]

12. (Extended) A solid cone has base radius 5 cm and perpendicular height 12 cm. Calculate:

(a) its slant height [2] (b) its total surface area, giving your answer in terms of π [2] (c) its volume, correct to 3 significant figures. [2]

13. (Extended) A solid cone has base radius 6 cm and height 12 cm. A smaller cone of height 4 cm is cut off the top, parallel to the base, leaving a frustum. Calculate the volume of the frustum, correct to 3 significant figures. [4]

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Answers

1. 1 m² = 10 000 cm² [1], so 3.4 m² = 3.4 × 10 000 = 34 000 cm² [1].

2. Area = 9 × 6 = 54 cm² [1]. Perimeter = 2(9 + 6) = 30 cm [1].

3. (a) C = 2πr = 2 × 3.142 × 8 = 50.3 cm (3 s.f.) [2]. (b) A = πr² = 3.142 × 8² = 201 cm² (3 s.f.) [2].

4. (a) Arc length = (60/360) × 2 × 3.142 × 8 = (1/6) × 50.27 [1] = 8.38 cm (3 s.f.) [1]. (b) Sector area = (60/360) × 3.142 × 8² = (1/6) × 201.1 [1] = 33.5 cm² (3 s.f.) [1].

5. (a) V = πr²h = 3.142 × 4² × 10 [1] = 503 cm³ (3 s.f.) [1]. (b) SA = 2πrh + 2πr² = (2 × 3.142 × 4 × 10) + (2 × 3.142 × 4²) [1] = 251.4 + 100.5 [1] = 352 cm² (3 s.f.) [1].

6. (a) Using Pythagoras: h² = 5² − 3² = 25 − 9 = 16 [1] → h = 4 cm [1]. (b) V = ⅓πr²h = ⅓ × 3.142 × 3² × 4 [1] = 37.7 cm³ (3 s.f.) [1].

7. Rectangle area = 10 × 6 = 60 cm² [1]. Semicircle radius = 3 cm, area = ½ × 3.142 × 3² = 14.14 cm² [1]. Remaining area = 60 − 14.14 = 45.9 cm² (3 s.f.) [1].

8. V = (4/3)πr³ = (4/3) × 3.142 × 6³ [1] = (4/3) × 3.142 × 216 = 905 cm³ (3 s.f.) [1].

9. SA = 4πr² → r² = 314.2 ÷ (4 × 3.142) [1] = 314.2 ÷ 12.568 = 25 [1] → r = √25 = 5 cm [1].

10. A = πr² = π × 6² [1] = 113.097… = 113 cm² [1].

11. (a) Arc length = (250/360) × 2 × π × 7 [1] = 30.54… = 30.5 cm [1]. (b) Area = (250/360) × π × 7² [1] = 106.90… = 107 cm² [1]. (c) Perimeter = arc + two radii = 30.54 + 7 + 7 = 44.5 cm [1]. Common mistake: leaving out the two radii.

12. (a) l² = 5² + 12² = 169 [1], so l = 13 cm [1]. (b) Curved surface = πrl = π × 5 × 13 = 65π; base = πr² = 25π [1]. Total = 90π cm² [1] (about 283 cm²). (c) V = ⅓πr²h = ⅓ × π × 25 × 12 = 100π [1] = 314 cm³ [1].

13. The small cone is similar to the large one, with height ratio 4 : 12 = 1 : 3, so its radius is 6 ÷ 3 = 2 cm [1]. Large cone: ⅓ × π × 6² × 12 = 144π [1]. Small cone: ⅓ × π × 2² × 4 = 16π/3 [1]. Frustum = 144π − 16π/3 = 416π/3 = 435.6… = 436 cm³ [1].

Indicative marking only: these are original questions, and the mark allocation is Marlbridge’s guide, not an official mark scheme.

Common mistake: using the diameter instead of the radius, or calculating 2πr (the circumference) instead of πr².

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Where marks are usually lost

• Converting between area units by the linear conversion factor instead of squaring it (1 m² is 10 000 cm², not 100 cm²).
• Forgetting the fraction of a full turn (angle ÷ 360) when finding an arc length or sector area.
• Mixing up the formulas for curved surface area and total surface area of a cylinder or cone — total surface area needs the circular end(s) added on.
• Using the slant height instead of the perpendicular height in the volume formula for a cone (they are different lengths, related by Pythagoras).
• Adding areas instead of subtracting when a compound shape has a piece removed, or the reverse.
• Rounding too early in a multi-step calculation: keep full calculator values in the working and round to 3 significant figures (or as instructed) only at the final answer.

Examiner report insight

• For a major sector or arc, first find the correct (larger) angle by subtracting the given angle from 360 degrees – applying the fraction-of-360 formula directly to the given (minor) angle answers the wrong region.
• Mixing up the radius and the diameter partway through a multi-step compound-shape or composite-solid calculation – write down which one the question actually gave before substituting.
• (Similarity, a related topic.) Finding a linear scale factor from a volume ratio requires a cube root, not a square root – the square root gives a linear scale factor from an area ratio instead.

Source: Cambridge International, 0580 Mathematics Principal Examiner Report for Teachers, June 2024 – Paper 21 Question 14 (major sector), Paper 13 Question 23 and Paper 21 Question 7(a) (diameter used instead of radius), Paper 22 Question 20 (volume scale factor). Paraphrased; checked against the report on 2026-09-27.

Approaching mensuration questions

The most reliable way to avoid unit and formula errors on this topic is to write down which formula the question needs before substituting any numbers, and to check the given units against what the answer needs to be in – since a wrong unit conversion or a swapped slant/perpendicular height produces an answer that still looks numerically plausible, it will not stand out as obviously wrong the way an arithmetic slip often does. For compound shapes, sketch (or describe to yourself) exactly which simple shapes the compound shape is built from and whether each one is added or removed, before calculating any individual area or volume.