Volume of a Sphere

Archimedes found the sphere's volume by fitting it in a cylinder: a ball that just fits inside a can, touching the top, the bottom and the sides. The ball fills exactly two thirds of the can.

That can has radius r and height 2r, so it holds $\pi\cdot {r}^{2}\cdot 2r = 2\cdot \pi\cdot {r}^{3}$. Two thirds of that is $\frac{4}{3}\cdot \pi\cdot {r}^{3}$. For this sphere of radius 3: $\frac{4}{3}\cdot \pi\cdot {3}^{3} = \frac{4}{3}\cdot \pi\cdot 27 = 36\cdot \pi$.

The rule is Volume of a sphere: $V = \frac{4}{3}\cdot \pi\cdot {r}^{3}$. This time the radius is cubed, because a sphere grows in all three directions at once.