Hi there, I have a homework question that asks why the volume of a pyramid is 1/3BH... I'm not even sure why it is.. I've just always been taught that formula and how to use it. Any ideas?
Let R be each side of the base and H is the height. Take a slice of the pyramid of thickness dh and length r at a height h from the vertex.
The volume of this slice is
dV = r^2*dh.By simple geometry you can see that R/H = r/h.
r = (R/H)*h.
dV = (R/H)^2*h^2*dh.
To find the total volume find the integration between h = 0 to h = H.
This is a cube ABCDEFGH, with lines joining all the vertices to a single vertex, A.
If you look very hard, you'll see that the cube is then made up of three identical pyramids, each with its vertex at A. The three pyramids have bases:
BFGC, EFGH and CGHDThe volume of each pyramid is therefore one-third the volume of the cube.
Now whatever pyramid you start with, you can always find a pyramid with the same base area and the same height that looks just like one of these three*. Its volume is therefore one-third base area x height.
* P.S. I don't think that's strictly true, because we've started with pyramids whose base area is equal to the square of their height. So, you may need to stretch (or compress) the pyramid a bit along its altitude. But the principle remains the same.