Why isn't the volume of a sphere: $\pi$$^\text{2}$$r^\text{3}$, instead it is $\frac{4}{3}$$\pi$$r^\text{3}$? Like wise the surface area is 4$\pi$$r^\text{2}$and not 2$\pi$$^\text{2}$$r^\text{2}$.
Simply take a 2D circle and rotate on same center and radius perpendicular circle and we get a sphere. But this isn't consistent among all the shapes which have a common axis.
I believe the only repeated/common things in this derivation are the pole of intersection and the axis.
Thanks for your help.