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