Proposition: Prop. 13.10: Square on Side of Regular Pentagon inscribed in Circle equals Squares on Sides of Hexagon and Decagon inscribed in sa

(Proposition 10 from Book 13 of Euclid's “Elements”)

If an equilateral pentagon is inscribed in a circle then the square on the side of the pentagon is (equal to) the (sum of the squares) on the (sides) of the hexagon and of the decagon inscribed in the same circle$^\dag$


Modern Formulation

If the circle is of unit radius then the side of the pentagon is \[\frac{\sqrt{10-2\,\sqrt{5}}}2.\]

Proofs: 1

Proofs: 1 2
Propositions: 3

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