The Frustum of a Pyramid. A frustum of a pyramid is the part of the pyramid included between its base and a plane parallel to its base. The figure below shows a square pyramid from which a smaller square pyramid has been cut off.

Each edge of the base of the large pyramid has length a, and its altitude, VO, has length x + h.

48) Write an expression for the volume of the large pyramid.

Each edge of the base of the small pyramid has a length b and its altitude, VP, has length x.
49) Write an expression for its volume.
50) Write an expression for the volume of the frustum (the part shown in green.)
Given that OM and PN are apothems of ABCD and EFGH, it can be shown that △VOM ~ △VPN.
51) Write experiments for OM and PN in terms of a and b.
52) Why is VO/VP = OM/PN?
53) Rewrite this proportion in terms of x, h, a, and b.
54) Solve your equation for x in terms of h, a, and b.
55) Substitute for x in your answer to exercise 50 and show that the volume of the frustum is ⅓ (a2 + ab + b2)h.

There is evidence that the Egyptians used this expression to find the volume of the frustum of a square based pyramid as early as 1890 B. C.*
56) Find the volume of such frustum for which a = 12, b = 3, and h = 8.

The ancient Babylonians used a different expression, ½(a2 + b2)h, to find the volume of the frustum of a square pyramid.

57) Is this expression also correct? Explain.