Volumes of Frustums

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A frustum is that past of a pyramid or cone that lines 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 volume of the original cone isand the volume of the smaller cone isThe difference between these two is the volume of the frustum (above bottom right), (2)

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


The volume of the frustum is seven eighths the volume of the cone.

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