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

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Theorem

In the words of Hypsicles of Alexandria:

(side of cube) : (side of icosahedron) = (content of dodecahedron) : (content of icosahedron)

(The Elements: Book $\text{XIV}$: Proposition $8$)


Lemma

In the words of Hypsicles of Alexandria:

If two straight lines be cut in extreme and mean ratio, the segments of both are in one and the same ratio.

(The Elements: Book $\text{XIV}$: Proposition $8$ : Lemma)


Proof

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

From Proposition $3$ of Book $\text{XIV} $: Circle Circumscribing Pentagon of Dodecahedron and Triangle of Icosahedron in Same Sphere:

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 Proposition $6$ of Book $\text{XII} $: Sizes of Pyramids of Same Height with Polygonal Bases are as Bases:

the ratios of these pyramids are to one another as their bases.

Thus:

\(\displaystyle 12 \text { pentagons} : 20 \text { triangles}\) \(=\) \(\displaystyle 12 \text { pyramids on pentagons} : 20 \text { pyramids on triangles}\)
\(\displaystyle \therefore \ \ \) \(\displaystyle \text {surface of dodecahedron} : \text {surface of icosahedron}\) \(=\) \(\displaystyle \text {volume of dodecahedron} : \text{volume of icosahedron}\)


Therefore from Proposition $6$ of Book $\text{XIV} $: Ratio of Sizes of Surfaces of Cube and Regular Icosahedron in Same Sphere:

\(\displaystyle \text {volume of dodecahedron} : \text{volume of icosahedron}\) \(=\) \(\displaystyle \text {side of cube} : \text {side of icosahedron}\)

$\blacksquare$


Historical Note

This theorem is Proposition $8$ of Book $\text{XIV}$ of Euclid's The Elements.


Sources