正多面体的体积与表面积之比Volume to Area ratio for Regular Solids

The volume of a sphere of radius r is

V = 4πr³ / 3

and the surface area is

A = 4πr²

and so the ratio of volume to area is

V / A = r / 3.

Surprisingly, the same ratio holds for all regular solids if r is the radius of the largest sphere that can be inscribed inside the regular solid.

For example, if the edge of a cube is a, then r = a/2. The volume is 8r³, the area is 24r², and the ratio is r/3.

The relationship between edge length and radius, and between radius and volume, is more complicated for the four other regular solids (tetrahedron, octahedron, dodecahedron, and icosahedron). However, in each case the ratio of volume to area is r/3.

The proof is surprisingly simple. Pick a face and form a pyramid by connecting each face vertex to the center of the inscribed sphere. The pyramid has height r and volume equal to B/3 where B is the area of the base. If the regular solid has f faces, the volume of the solid is fBr / 3 and the area is fB. So the ratio of volume to area is r/3.

The theorem generalizes to n > 3 dimensions. The formula for the volume of a pyramid in n dimensions is Bh/n where B is the (n − 1)-dimensional volume of the base, and so the ratio of n-dimensional volume of a regular solid to (n − 1)-dimensional volume of its boundary is r/n.