Therefore, the formula for the volume of the sphere can be derived by writing an expression that represents the volume of

A sphere of radius r can be compared with a cylinder of radius r and height 2r. The two cones inside that cylinder have a combined volume of 2(⅓πr²·r) = ⅔πr³, while the cylinder’s volume is 2πr³. The volume between the cones and the cylinder is therefore 2πr³ − ⅔πr³ = ⁴⁄₃πr³, the sphere’s volume. So the expression represents the solid between the two cones and the cylinder.

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