Cones, spheres and pyramids · Grade 9

In Grade 9 you learn: surface area of cones and spheres.

Pyramids, cones and spheres are solids that do not have the same cross-section all the way through. A pyramid or cone holds exactly one third of the prism or cylinder with the same base and height, and a sphere has its own formulas for volume and surface area.

Key facts

  • Pyramid: V = ⅓ × base area × height.
  • Cone: V = ⅓πr²h; curved surface area = πrl, where l is the slant height.
  • Slant height: l² = r² + h² (Pythagoras).
  • Sphere: V = 4/3 πr³; surface area = 4πr².
  • Hemisphere: half a sphere; its total surface area is 2πr² + πr² = 3πr².

Worked examples

Example 1

Find the volume of a cone with radius 5 cm and height 12 cm, to 1 d.p.

Answer: 314.2 cm³

Example 2

The same cone (r = 5 cm, h = 12 cm): find its curved surface area, to 1 d.p.

Answer: 204.2 cm²

Example 3

A metal sphere of radius 6 cm is melted and made into spheres of radius 2 cm. How many small spheres are made?

Answer: 27

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