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.


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

  1. Lemma: Lem. 13.18: Angle of the Pentagon

Definitions: 1 2 3 4 5

Thank you to the contributors under CC BY-SA 4.0!



Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

  1. Prime.mover and others: "Pr∞fWiki",, 2016