◀ ▲ ▶Branches / Geometry / Euclideangeometry / Elementseuclid / Book13platonicsolids / 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: 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$^\dag$
* Let $ABCDE$ be a circle.
* And let the equilateral pentagon $ABCDE$ have been inscribed in circle $ABCDE$.
* I say that the square on the side of pentagon $ABCDE$ is the (sum of the squares) on the sides of the hexagon and of the decagon inscribed in circle $ABCDE$.
Modern Formulation
If the circle is of unit radius then the side of the pentagon is \[\frac{\sqrt{102\,\sqrt{5}}}2.\]
Table of Contents
Proofs: 1
Mentioned in:
Proofs: 1 2
Propositions: 3
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References
Adapted from (subject to copyright, with kind permission)
 Fitzpatrick, Richard: Euclid's "Elements of Geometry"
Adapted from CC BYSA 3.0 Sources:
 Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016