Cones, spheres and pyramids

Pyramids and cones come to a point; spheres are perfectly round. Their volumes use special formulas: a cone or pyramid is a third of the prism or cylinder around it.

Key facts

  • Pyramid: V = ⅓ × base area × height.
  • Cone: V = ⅓πr²h; curved surface = πrl (l = slant height).
  • Sphere: V = 4/3 πr³; surface area = 4πr².
  • Find the slant height with Pythagoras: l² = r² + h².

Worked examples

Example 1

Find the volume of a cone with radius 3 cm and height 4 cm, in terms of π. Then find its total surface area.

  1. V = ⅓ × π × 3² × 4 = 12π
  2. Slant height: √(3² + 4²) = 5.
  3. Curved: π × 3 × 5 = 15π. Base: π × 3² = 9π. SA = 24π cm²

Answer: V = 12π cm³, SA = 24π cm²

Example 2

A sphere has volume 288π cm³. Find its radius.

  1. 4/3 πr³ = 288π
  2. r³ = 216 r = 6 cm

Answer: 6 cm