Surface area

Surface area is the total area of all the faces of a solid, measured in square units such as cm². Unfold the solid into its net, find the area of each face, and add them up.

Key facts

  • Surface area: add the areas of all the faces (a net helps).
  • A cube has 6 equal square faces; a cuboid has 3 pairs of matching rectangles.
  • A prism's surface: two ends plus a rectangle for each side.
  • Cylinder: curved surface = 2πrh; total surface = 2πr² + 2πrh.
  • Sphere: surface area = 4πr².

Worked examples

Example 1

Find the surface area of a cube with edges of 3 cm.

  1. Each face is a 3 cm square. 3 × 3 = 9 cm²
  2. A cube has 6 faces. 6 × 9 = 54 cm²

Answer: 54 cm²

Example 2

Find the surface area of a cuboid 5 cm by 4 cm by 2 cm.

  1. A cuboid has three pairs of matching faces. 5 × 4 = 20, 5 × 2 = 10, 4 × 2 = 8
  2. Add one of each, then double. 2 × (20 + 10 + 8) = 76 cm²

Answer: 76 cm²

Example 3

A cuboid is 6 cm long, 5 cm deep and 4 cm tall. Find its volume and its surface area.

  1. Fill the front with 1 cm cubes: 6 across and 4 up. 6 × 4 = 24 cubes
  2. The box is 5 cm deep, so it holds 5 slices like that. 24 × 5 = 120 cm³

Answer: Volume 120 cm³, surface area 148 cm²

Example 4

A cube has a surface area of 150 cm². How long is each edge?

  1. A cube has 6 equal square faces. 150 ÷ 6 = 25 cm²
  2. Each face is a square with an area of 25 cm². √25 = 5 cm

Answer: 5 cm

Example 5

A triangular prism is 10 cm long. Its ends are right-angled triangles with sides of 3 cm, 4 cm and 5 cm. Find its surface area.

  1. The two triangular ends: 2 × ½ × 3 × 4 = 12 cm²
  2. The three rectangles: 3 × 10 + 4 × 10 + 5 × 10 = 120 cm²
  3. Add them. 12 + 120 = 132 cm²

Answer: 132 cm²

Example 6

Find the curved surface area of a cylinder with radius 4 cm and height 6 cm, to 1 decimal place.

  1. Unrolled, the curved surface is a rectangle: the circumference by the height. 2 × π × 4 × 6 = 48π
  2. Round to 1 decimal place. 48π = 150.796… ≈ 150.8 cm²

Answer: 150.8 cm²

Example 7

Find the total surface area of a cylinder with radius 2 cm and height 5 cm, to 1 decimal place.

  1. The two ends. 2 × π × 2² = 8π
  2. The curved surface unrolls into a rectangle. 2π × 2 × 5 = 20π
  3. Add them. 28π = 88.0 cm²

Answer: 88.0 cm²

Example 8

Find the surface area of a sphere with radius 5 cm, to 1 decimal place.

  1. Square the radius, then multiply by 4π. 4 × π × 5² = 100π
  2. Round to 1 decimal place. 100π = 314.159… ≈ 314.2 cm²

Answer: 314.2 cm²