Volume to Area ratio for Regular Solids

The volume of a sphere of radius r is

V = 4r^3 / 3

and the surface area is

A = 4r^2

and so the ratio of volume to area is

V / A = r / 3.

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

For example, if the edge of a cube isa, thenr =a/2. The volume is 8r^3, the area is 24r^2, and the ratio isr/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 isr/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 toB/3 whereB is the area of the base. If the regular solid hasf faces, the volume of the solid isfBr / 3 and the area isfB. So the ratio of volume to area isr/3.

The theorem generalizes to n > 3 dimensions. The formula for the volume of a pyramid in n dimensions isBh/n whereB 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.

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