Suppose that V(r) is the ball volume dependent on the radius. Then dV(r) = V'(r)dr is the volume of a thin sphere enveloping the ball whose thickness is dr. On the other hand, the volume of this sphere is S(r)dr where S(r) is the surface area. Therefore, S(r) = V'(r).
Now, suppose that for an -dimensional ball, for some constant . Then and the solution to the equation is .