I read in a book a few months ago that the volume of a (solid) pyramid with a base that is ANY polygon (I'm not sure if it mentioned it being regular or not) is equal to
$$\frac{1}{3}\times A\times h$$
where $A$ is the area of the base (i.e. of the polygon) and $h$ is the height of the pyramid.
This seems to be true in many cases, such as when the base is a square, a triangle and when the base is a circle (ie the pyramid becomes a cone).
My question is, how can we prove this? I simply have no idea.
Thank you for your help.
Best Answer
Any polygon can be decomposed into triangles, giving $A = A_1 + \cdots + A_n$.
Thus, $$\begin{align} V &= V_1 + \cdots + V_n\\ &=\frac13A_1h + \cdots + \frac13A_nh\\ &=\frac13(A_1 + \cdots + A_n)h\\ &=\frac13Ah. \end{align}$$
For the base case, you can decompose a prism like so. To see that the three pyramids have the same volume, note that the left and middle pyramids share a blue base and have the same height; similarly, the middle and right pyramids share a red base and have the same height.