Ratio of Volumes of Regular Dodecahedron and Regular Icosahedron in Same Sphere

Proof
Let a regular dodecahedron, a regular icosahedron and a cube be inscribed in a given sphere.

From :
 * the circle which circumscribes the regular pentagon which is the face of the regular dodecahedron is the same size as the circle which circumscribes the equilateral triangle which is the face of the regular icosahedron.

In a sphere, equal sections are equally distant from the center.

Thus the perpendiculars from the center to the faces of the regular icosahedron and regular dodecahedron are equal.

So the pyramids whose apices are the center of the sphere and whose bases are the faces of these polyhedra are of the same height.

Therefore from :
 * the ratios of these pyramids are to one another as their bases.

Thus:

Therefore from :