Proposition: Prop. 13.11: Side of Regular Pentagon inscribed in Circle with Rational Diameter is Minor

(Proposition 11 from Book 13 of Euclid's “Elements”)

If an equilateral pentagon is inscribed in a circle which has a rational diameter then the side of the pentagon is that irrational (straight line) called minor.

Modern Formulation

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

However, this length can be written in the minor. \[\sqrt{5}\sqrt{\left(1+\frac{\rho}{\sqrt{1+\rho^2}}\right)\frac 12} - \sqrt{\left(1-\frac{\rho}{\sqrt{1+\rho^2}}\right)\frac 12}, \]

with $\rho=2$.

Proofs: 1

Proofs: 1

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