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MathidiaMathematics Academy
Circles, sectors and solids
Lesson 58 of 75
Standard practice

Cones and spheres

Objective. Use formulae for curved solids and justify dimensions.

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  2. 2. Worked example
  3. 3. Practice
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Learn: the key idea

Cones and spheres are curved solids with standard volume and surface area formulae that must be recalled and applied with correct substitution. A cone of radius r and height h has volume (1/3)πr²h and curved surface area πrl (l is the slant height), while a sphere of radius r has volume (4/3)πr³ and surface area 4πr². For example, a cone with r = 3 cm and h = 4 cm has slant height l = √(3²+4²) = 5 cm, giving curved surface area 15π cm². Always find the slant height using Pythagoras first if only the vertical height is given.

Worked example

A sphere has radius 6 cm. Calculate its volume, leaving your answer in terms of π, for this question on cones and spheres.

Diagram of a circle
  1. 1Volume formula for a sphere: V = (4/3)πr³
  2. 2r = 6, so r³ = 216
  3. 3V = (4/3)π(216)
  4. 4V = 288π cm³

Answer: 288π cm³

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Common mistake. Students often confuse the cone's height with its slant height when using the surface area formula πrl; remember l is the slanted distance from the apex to the base edge, found via Pythagoras if needed.

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