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Truncated Square Pyramid

The truncated square pyramid is a special case of a Pyramidal Frustum for a Square Pyramid. Let the base and top side lengths of the truncated pyramid be $a$ and $b$, and let the height be $h$. Then the Volume of the solid is

V={\textstyle{1\over 3}}(a^2+ab+b^2)h.

This Formula was known to the Egyptians ca. 1850 BC. The Egyptians cannot have proved it without calculus, however, since Dehn showed in 1900 that no proof of this equation exists which does not rely on the concept of continuity (and therefore some form of Integration).

See also Frustum, Pyramid, Pyramidal Frustum, Square Pyramid

© 1996-9 Eric W. Weisstein