Lune (geometry)

In the 5th century BC, Hippocrates of Chios showed that the Lune of Hippocrates and two other lunes could be exactly squared (converted into a square having the same area) by straightedge and compass. In 1766 the Finnish mathematician Daniel Wijnquist, quoting Daniel Bernoulli, listed all five geometrical squareable lunes, adding to those known by Hippocrates. In 1771 Leonard Euler gave a general approach and obtained certain equation to the problem. In 1933 and 1947 it was proven by Nikolai Chebotaryov and his student Anatoly Dorodnov that these five are the only squarable lunes.[3][1]

The area of a lune formed by circles of radii a and b (b>a) with distance c between their centers is[3]

${\displaystyle A=2\Delta +a^{2}\sec ^{-1}\left({\frac {2ac}{b^{2}-a^{2}-c^{2}}}\right)-b^{2}\sec ^{-1}\left({\frac {2bc}{b^{2}+c^{2}-a^{2}}}\right),}$

where ${\displaystyle {\text{sec}}^{-1}}$ is the inverse function of the secant function, and where

${\displaystyle \Delta ={\frac {1}{4}}{\sqrt {(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}}}$

is the area of a triangle with sides a, b and c.

• Arbelos
• Crescent
• Gauss–Bonnet theorem
• Lens

• The Five Squarable Lunes at MathPages