Derivation:

The volume of a sphere can be described by a number of pyramids with n-gonal bases that completely cover the sphere, with their vertices at the center of the sphere. The volume of the sphere is then

V = *n*×(1/3)*b**r*

where *n* is the number of pyramids, *b* is the area of one of the pyramid's bases, and *r* is the radius of the sphere. This equation can be rearranged to read:

V = *n*×*b*(1/3)*r*

But what is *n*×*b* equal to? The surface area of the sphere! Thus, we can write:

V = SA×(1/3)*r*

where SA is surface area.

Now it's time to start solving.

(4/3)π*r*³ = SA(1/3)*r*

(4/3)π*r*² = (1/3)SA

SA = 4π*r*²

Q.E.D.

Of course, if you know the *calculus*, or are a *smartass*, or both, then you could just show that dV/d*r* = SA, and ∫A d*r* = V. But where's the fun in that?