Ratio of Sizes of Surfaces of Regular Dodecahedron and Regular Icosahedron in Same Sphere

Proof

 * Euclid-XIV-3.png $\qquad$ Euclid-XIV-4.png

What is meant by its perpendicular is a perpendicular from the center of the circle circumscribed around it to the side of it.

In the above diagram:

From :
 * the surface of the dodecahedron is $30$ times its perpendicular times its side.

From :
 * the surface of the icosahedron is $30$ times its perpendicular times its side.

Hence the result.