Proposition: Prop. 13.18: There are only Five Platonic Solids

(Proposition 18 from Book 13 of Euclid's “Elements”)

To set out the sides of the five (aforementioned) figures, and to compare (them) with one another.

$AB$ is the diameter of the sphere.
$AF$ is equal to the side of the pyramid.
$BF$ is the side of the cube.
$MB$ is thus (the side) of the pentagon (inscribed in the circle) [Prop. 13.10], [Prop. 1.47] ... And (the side) of the pentagon is (the side) of the icosahedron [Prop. 13.16].
$NB$ is the side of the dodecahedron [Prop. 13.17 corr.] .
And the (square) on (the side) of the octahedron is one and a half times the square on (the side) of the cube.
So, I say that, beside the five aforementioned figures, no other (solid) figure can be constructed (which is) contained by equilateral and equiangular (planes), equal to one another.

fig18e

Modern Formulation

If the radius of the given sphere is unity then the sides of pyramid (i.e., tetrahedron), octahedron, cube, icosahedron, and dodecahedron, respectively, satisfy the following inequality: \[\sqrt{\frac 83} > \sqrt{2} > \sqrt{\frac 43} > \frac{\sqrt{10 -2\,\sqrt{5}}}{\sqrt{5}} > \frac{\sqrt{15}-\sqrt{3}}3.\]

Moreover, these five Platonic solids (i.e. solids contained by equilateral and equiangular rectilinear figures) are the only existing ones.

Proofs: 1