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Volume and surface area of 3D shapes: formulas and examples

Find the volume and surface area of cuboids, prisms, cylinders, cones and spheres, with worked examples and a quick formula list.

IntermediateUpdated 5 Nov 2026
GCSEHigh School

Curriculum fit: GCSE Maths Geometry and measures (Foundation and Higher; cones and spheres mainly Higher). US: Geometry (Geometric Measurement and Dimension).

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What is it?

Volume measures how much space a 3D shape fills and is measured in cubic units such as cm³. Surface area is the total area of all the faces and is measured in square units.

For a prism, volume is the area of the cross-section multiplied by the length. For curved shapes you use formulas that include π.

Key formulas and ideas

  • Cuboid: V = lwh. Surface area = 2(lw + lh + wh).
  • Prism: V = cross-section area × length.
  • Cylinder: V = πr²h. Surface area = 2πr² + 2πrh.
  • Cone: V = ⅓πr²h. Sphere: V = (4/3)πr³ and surface area = 4πr².

Worked examples

Example 1: Cylinder

Question: Find the volume of a cylinder with radius 3 cm and height 10 cm.

  1. V = πr²h.
  2. = π × 3² × 10 = 90π.
  3. 90π ≈ 282.7.

Answer: 90π ≈ 282.7 cm³

Example 2: Cuboid surface area

Question: Find the surface area of a cuboid 4 cm by 5 cm by 6 cm.

  1. Face areas: lw = 20, lh = 24, wh = 30.
  2. Surface area = 2 × (20 + 24 + 30) = 2 × 74.

Answer: 148 cm²

Example 3: Sphere

Question: Find the volume of a sphere with radius 3 cm.

  1. V = (4/3)πr³ = (4/3) × π × 27.
  2. = 36π ≈ 113.1.

Answer: 36π ≈ 113.1 cm³

Common mistakes

  • Using diameter instead of radius.
  • Forgetting to cube the radius for a sphere.
  • Mixing cm² and cm³.
  • Leaving out the circles when finding the surface area of a closed cylinder.

Practice questions

1. Volume of a cuboid 2 cm by 3 cm by 4 cm?
2. Volume of a cylinder with radius 2 cm and height 5 cm (to 1 d.p.)?
3. A triangular prism has cross-section area 12 cm² and length 5 cm. Its volume is…
4. Surface area of a cube with side 3 cm?
5. Volume of a sphere with radius 1 (in terms of π)?

Frequently asked questions

How do I convert between cm³ and litres?

1000 cm³ = 1 litre.

Which formulas are given in the exam?

This varies by exam board and tier. Check the formulae list in your specification.

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Sources and review

Reviewed by StudyMorrow editorial review on 5 Nov 2026.