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)br
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)πr³ = SA(1/3)r
(4/3)πr² = (1/3)SA
SA = 4πr²
Of course, if you know the calculus, or are a smartass, or both, then you could just show that dV/dr = SA, and ∫A dr = V. But where's the fun in that?