Shown in Fig. 1 is a pile of cylinders that approximates one half of a sphere.

Fig. 1

From Fig. 1, we see .

If denotes the volume of cylinder in the pile, then

.

Let

,

we have

see “Little Bird and a Recursive Generator“

.

As

Therefore, gives

Similarly, we approximate a cone

by a stack of cylinders:

Fig. 2

Since

and

,

we have

If denotes the cylinder, then

.

Let

we have

.

Since as , , we have

We now derive the formula for a sphere’s surface area.

Let us approximate a sphere by many cones:

Fig. 3

From Fig. 3, we see that as ,

.

Consequently,

and,

.

It follows that

Hence,