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In plane geometry, thecrescent shape formed by two intersecting circles is called alune. In each diagram, two lunes are present, and one is shaded in grey.

Inplane geometry, alune (from Latin luna 'moon') is the concave-convex region bounded by two circulararcs.^[1] It has one boundary portion for which the connecting segment of any two nearby points moves outside the region and another boundary portion for which the connecting segment of any two nearby points lies entirely inside the region. A convex-convex region is termed alens.^[2]

Formally, a lune is therelative complement of onedisk in another (where they intersect but neither is a subset of the other). Alternatively, if${\displaystyle A}$ and${\displaystyle B}$ are disks, then${\displaystyle A\setminus A\cap B}$ is a lune.

In the 5th century BC,Hippocrates of Chios showed that theLune of Hippocrates and two other lunes could beexactly squared (converted into a square having the same area) bystraightedge and compass. Around 1000,Alhazen attempted to square a circle using a pairof lunes now bearing his name. In 1766 the Finnish mathematician Daniel Wijnquist, quotingDaniel Bernoulli, listed all five geometrical squareable lunes, adding to those known by Hippocrates. In 1771Leonhard Euler gave a general approach and obtained a certain equation to the problem. In 1933 and 1947 it was proven byNikolai 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 radiia andb (b>a) with distancec between their centers is^[3]

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

where${\displaystyle {\text{cos}}^{-1}}$ is the inverse cosine or arccosine, and where

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

is thearea of a triangle with sidesa, b andc.

• Arbelos
• Crescent
• Gauss–Bonnet theorem
• Lens

• The Five Squarable Lunes at MathPages

• Piecewise-circular curves

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