Question

How is the base of a shape related to its volume?

Answer 1

If prism-like then the volume is `"base" xx "height"`

If pyramid-like then the volume is `1/3 xx "base" xx "height"`

Explanation:

If a solid is prism-like, with top the same shape and size as the bottom, then its volume is the product of its base area and height.

If a solid is pyramid-like, with the top being a single point and sloping sides, then its volume is one third of the product of its base area and height.

One way you can see this starts by dividing a cube into six square base pyramids with common apex in the centre of the cube.

If the length of each side of the cube is `1`, then the base area of each pyramid is `1xx1=1`, its height `1/2` and its volume is `1/6`, being `1/6` of the volume of the `1xx1xx1` cube.

So such a square based pyramid has volume `1/3` of the product of its base area and height.

The formula remains true if we stretch or compress the pyramid uniformly in any one direction. The volume also remains constant if we subject the pyramid to a shear, retaining its height and base (think of a stack of coins).

Given any two dimensional shape, we can approximate it arbitrarily closely with a grid of squares, then make square based pyramids on each of those squares up to a common apex. The total volume is the sum of the volumes of the pyramids, which works out the same as `1/3` of the height multiplied by the sum of the areas of the squares. Since the squares approximate the area of the original shape, the volume approximated by the pyramids approximates the volume of a pyramid based on the original shape.