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0580/41 Oct/Nov 2025, Question 14: Volume, density and surface area

Cambridge IGCSE Mathematics 0580 · Paper 41 (Extended) · Calculator · 10 marks

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The question, in short

A metal sphere fits exactly inside a cube of side 8 cm. Work with its volume, the empty space, its mass, and a cylinder made from it.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (a) · 1 mark

Show that the volume of the sphere is 256π/3 cm³.

  1. 1The sphere touches all faces, so radius = 8 ÷ 2 = 4 cm
  2. 2Volume = 4/3 × π × 4³ = 256/3 π cm³

Answer4/3 × π × 4³ = 256/3 π

Part (b) · 3 marks

Find the percentage of the cube not filled by the sphere.

  1. 1Cube volume = 8³ = 512 cm³, sphere = 256/3 π = 268.08… cm³
  2. 2Not occupied = 512 − 268.08… = 243.91… cm³
  3. 3243.91… ÷ 512 × 100 = 47.64… ≈ 47.6%

Answer47.6 or 47.63 to 47.64…

Part (c) · 2 marks

The metal has density 7.86 g/cm³. Find the mass of the sphere in kilograms.

  1. 1The sphere touches every face, so its diameter is 8 cm and radius 4 cm.
  2. 2Volume = 4/3 × π × 4³ = 256π/3 cm³
  3. 3Mass = volume × density = 256π/3 × 7.86 = 2107.1… g
  4. 4Divide by 1000 for kg: 2.107… ≈ 2.11 kg

Answer2.11 or 2.107…

Part (d) · 4 marks

The sphere is melted into a solid cylinder of radius 3.1 cm. Find the cylinder's total surface area.

  1. 1π × 3.1² × h = 256/3 π, so h = 256/3 ÷ 9.61 = 8.879… cm
  2. 2Two ends = 2 × π × 3.1² = 60.38… cm²
  3. 3Curved surface = 2 × π × 3.1 × 8.879… = 172.95… cm²
  4. 4Total = 60.38… + 172.95… = 233.3… ≈ 233 cm²

Answer233 or 233.3 to 233.4

Stuck on a step? Read the free lesson: Surface Area of a Cuboid and Cylinder

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Also searched as: 0580/41/O/N/25 · 0580_w25_qp_41 · 0580 w25 41 · question 14.
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