Estimate the Volume of a Pyramid and Prism

Which has a greater volume, a prism or a pyramid, if the two have the same base and height? To find out, compare prisms and pyramids that have congruent bases and the same height.

Here is a base for a triangular prism and a triangular pyramid. Both figures have the same height. Compare the two figures. Which one appears to have a greater volume?

The prism may appear to be greater in volume. But how can you prove that the volume of the prism is greater than the volume of the pyramid? Put one figure inside of the other. The figure that is smaller will fit inside of the other figure.

This is shown in the diagram on the above. Both figures have congruent bases and the same height. The pyramid clearly fits inside of the prism. So the volume of the pyramid must be smaller.

Example 4

Show that the volume of a square prism is greater than the volume of a square pyramid.

Draw or make a square prism and a square pyramid that have congruent bases and the same height.

Now place the one figure inside of the other. The pyramid fits inside of the prism. So when two figures have the same height and the same base, the prism’s volume is greater.

In general, when you compare two figures that have congruent bases and are equal in height, the prism will have a greater volume than the pyramid.

The reason should be obvious. At the “bottom,” both figures start out the same—with a square base. But the pyramid quickly slants inward, “cutting away” large amounts of material while the prism does not slant.

Find the Volume of a Pyramid and Prism

Given the figure above, in which a square pyramid is placed inside of a square prism, we now ask: how many of these pyramids would fit inside of the prism?

To find out, obtain a square prism and square pyramid that are both hollow, both have no bottom, and both have the same height and congruent bases.

Now turn the figures upside down. Fill the pyramid with liquid. How many full pyramids of liquid will fill the prism up to the top?

In fact, it takes exactly three full pyramids to fill the prism. Since the volume of the prism is:

where stands for the area of the base and is the height of the prism, we can write:

or:

And, since the volume of a square prism is , we can write:

This can be written as the Volume Postulate for pyramids.

Volume of a Pyramid

Given a right pyramid with a base that has area and height :

Example 5

Find the volume of a pyramid with a right triangle base with sides that measure , , and . The height of the pyramid is .

First find the area of the base. The longest of the three sides that measure , , and must be the hypotenuse, so the two shorter sides are the legs of the right triangle.

Now use the postulate for the volume of a pyramid.

Example 6

Find the altitude of a pyramid with a regular pentagonal base. The figure has an apothem of , sides, and a volume of .

First find the area of the base.

Now use the value for the area of the base and the postulate to solve for .