Volume of Sphere from Surface Area

Theorem
The volume $V$ of a sphere of radius $r$ is given by:
 * $V = \dfrac {r A} 3$

where $A$ is the surface area of the sphere.

Proof
Let the surface of the sphere of radius $r$ be divided into many small areas.

If they are made small enough, they can be approximated to plane figures.

Let the areas of these plane figures be denoted:
 * $a_1, a_2, a_3, \ldots$

Let the sphere of radius $r$ be divided into as many pyramids whose apices are at the center and whose bases are these areas.

From Volume of Pyramid, their volumes are:
 * $\dfrac {r a_1} 3, \dfrac {r a_2} 3, \dfrac {r a_3} 3, \ldots$

The volume $\mathcal V$ of the sphere is given by the sum of the volumes of each of these pyramids:

But $a_1 + a_2 + a_3 + \cdots$ is the surface area of the sphere.

Hence:

It needs to be noted that this proof is intuitive and non-rigorous.

Historical Note
This was the method used by as an offshoot of the proof he gave for the area of a circle