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 Question from Darah, a student: Suppose you want to build a toy pyramid with a square base and a volume of 9000 cubic centimeters. What are the dimensions of a pyramid you might design? How many such pyramid designs are there?

Hi Darah.

Technically, a "pyramid" doesn't have to have the vertex (top point) directly above the center of the base. But let's just think about those pyramids with the vertex directly up from the center of the base - these are called "right pyramids".

If you have a really big square base, you don't have to go very high up to have a big volume. But if you shrink the base, you have to go higher up with the vertex to have the same volume. But the point is that you CAN have the same volume when the size of the bases is different. In fact, for any size base, there is a height you can choose that will make the volume exactly what you want. So there are an infinite number of right pyramids that you can create which have a square base and a volume of 9000 cubic centimeters.

So pick a base size, then use that to calculate the necessary height for the given volume. You need to know that the volume of a pyramid (whether a right pyramid or not) is (1/3) A h, where A is the area of the base.

Hope this helps,
Stephen La Rocque.

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.