jmc

Question

A wooden model of a square pyramid has a base edge of 12 cm and an altitude of 8 cm. A cut is made parallel to the base of the pyramid that separates it into two pieces: a smaller pyramid and a frustum. Each base edge of the smaller pyramid is 6 cm and its altitude is 4 cm. How many cubic centimeters are in the volume of the frustum?

Accepted Answer

The piece that is removed from the original pyramid to create the frustum is itself a square pyramid that is similar to the original pyramid. The ratio of corresponding side lengths is 1/2, so the piece that was removed has volume

(1/2)^{3} = 1/8

of the volume of the original pyramid. Therefore, the remaining frustum has volume

1 - (1/8) = 7/8

of the original pyramid.

The original pyramid has base area

1 2^{2} = 144

square cm, so its volume is

144 \cdot 8/3 = 48 \cdot 8

cubic centimeters. Therefore, the frustum has volume

\frac{7}{8} \cdot (48 \cdot 8) = 48 \cdot 7 = 336 cubic centimeters .