Question

The volume enclosed by a sphere of radius r is (4/3)πr^3. The surface area of the same sphere is 4πr^2. You may already have noticed that the volume is exactly (1/3)r times the surface area. Explain why this relationship should be expected. One way is to consider a billion-faceted polyhedron that is circumscribed about a sphere of radius r; how are its volume and surface area related?

Answer 1

The volume of a pyramid is 1/3 Bh
Consider the many-faceted polyhedron as a collection of pyramids.
Each small patch -- of area, say, B, of the surface of the sphere is the base of a pyramid of height r, giving it a volume of 1/3 Br.

Adding up all the patches, you get the surface of the sphere, multiplied by 1/3 r, giving the volume of the sphere.