Question

What is the volume `v` of a sphere of radius `r`?

Answer 1

`V=4/3pir^3`

Answer 2

`v = 4/3 pir^3`

Explanation:

The surface area of a sphere of radius `r` is `4pi r^2`

Imagine dissecting a sphere into a large number of slender pyramids, with apices at the centre and (slightly rounded) bases tesselating the surface. As you use more pyramids, the bases get flatter.

The volume of each pyramid is `1/3 xx "base" xx "height"`

So the total volume of all the pyramids is:

`v = sum 1/3 xx "base" xx "height" = r/3 sum "base" = r/3 * 4pir^2 = 4/3 pir^3`