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Lune


Lune

A lune is a plane figure bounded by two circular arcs of unequal radii, i.e., a crescent. (By contrast, a plane figure bounded by two circular arcs of equal radius is known as a lens.) For circles of radius a and b>a whose centers are separated by a distance c, the area of the lune is given by

 A=a^2[tan^(-1)((a^2-b^2+c^2)/(4Delta))+cos^(-1)((b-c)/a)+tan^(-1)((b-c)/(sqrt((a+b-c)(a-b+c))))] 
 -b^2[tan^(-1)((a^2-b^2-c^2)/(4Delta))+pi/2]+2Delta  
=2Delta+a^2sec^(-1)((2ac)/(b^2-a^2-c^2))-b^2sec^(-1)((2bc)/(b^2+c^2-a^2)),
(1)

where

 Delta=1/4sqrt((a+b+c)(b+c-a)(c+a-b)(a+b-c))
(2)

is the area of the triangle with side lengths a, b, and c. The second of these can be obtained directly by subtracting the areas of the two half-lenses whose difference producing the colored region above.

Lunes

In each of the figures above, the area of the lune is equal to the area of the indicated triangle. Hippocrates of Chios squared the above left lune (Dunham 1990, pp. 19-20; Wells 1991, pp. 143-144), as well as two others, in the fifth century BC. Two more squarable lunes were found by T. Clausen in the 19th century (Shenitzer and Steprans 1994; Dunham 1990 attributes these discoveries to Euler in 1771). In the 20th century, N. G. Tschebatorew and A. W. Dorodnow proved that these are the only five squarable lunes (Shenitzer and Steprans 1994).

DoubleLune

Hippocrates also proved that, in the figure above, the sum of the areas of the two colored lunes is equal to the area of the triangle (Pappas 1989, pp. 72-73). Individually, the area of the lunes are

A_(top)=1/8[a(2b+api)-2(a^2+b^2)tan^(-1)(a/b)]
(3)
A_(left)=1/8[a(2b-api)+2(a^2+b^2)tan^(-1)(a/b)],
(4)

where the lengths of the horizontal and vertical legs of the triangle are a and b, respectively.


See also

Annulus, Arc, Circle, Circle-Circle Intersection, Flower of Life, Lens, Mohammed Sign, Oval, Salinon, Semicircle, Spherical Lune

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References

Dunham, W. "Hippocrates' Quadrature of the Lune." Ch. 1 in Journey through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 1-20, 1990.Heath, T. L. A History of Greek Mathematics, Vol. 1: From Thales to Euclid. New York: Dover, p. 185, 1981.Pappas, T. "Lunes." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 72-73, 1989.Shenitzer, A. and Steprans, J. "The Evolution of Integration." Amer. Math. Monthly 101, 66-72, 1994.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 143-144, 1991.

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Lune

Cite this as:

Weisstein, Eric W. "Lune." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Lune.html

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