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

Theorem

In the words of Hypsicles of Alexandria:

(surface of dodecahedron) : (surface of icosahedron) = (side of pentagon) . (its perpendicular) : (side of triangle) . (its perp.).

(The Elements: Book $\text{XIV}$: Proposition $5$)

Proof

$\qquad$

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 Proposition $3$ of Book $\text{XIV} $: Size of Surface of Regular Dodecahedron:

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

From Proposition $4$ of Book $\text{XIV} $: Size of Surface of Regular Icosahedron:

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

Hence the result.

$\blacksquare$

Historical Note

This proof is Proposition $5$ of Book $\text{XIV}$ of Euclid's The Elements.

Sources

• 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): The So-Called Book $\text{XIV}$, by Hypsicles