Surface Areas of Frustums

A frustum is that past of a pyramid or cone that lies between two parallel planes cutting it. If one of the planes is the base and the other is horizontal then the result is the shape shown below right (for a cone).

The smaller cone top right and the original cone left are similar shapes, so(1)

The curved surface area of the original cone isand the curved surface area of the smaller cone isThe difference between these two is the curved surface area of the frustum(above bottom right), (2)

Rearranging (1) to giveand substituting into (2) gives

Add on the top and bottom to give