A lune is the region inside one disk and outside another when their boundary circles intersect in two distinct points. It is bounded by two circular arcs. The radii may be equal or unequal. By contrast, the common region of the two disks is a lens.
For circles of radii 
and 
whose centers are separated by a distance 
with 
,
the area of the part of the smaller disk
outside the larger 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^2cos^(-1)((b^2-a^2-c^2)/(2ac))-b^2cos^(-1)((b^2+c^2-a^2)/(2bc)),](/images/equations/Lune/NumberedEquation1.svg) |
(1)
|
where
 |
(2)
|
is the area of the triangle with side lengths 
, 
,
and 
. The second of these can be obtained
directly by subtracting the areas of the two half-lenses
whose difference producing the colored region above.
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).
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
 |  | ![1/8[a(2b+api)-2(a^2+b^2)tan^(-1)(a/b)]](/images/equations/Lune/Inline10.svg) |
(3)
|
 |  | ![1/8[a(2b-api)+2(a^2+b^2)tan^(-1)(a/b)],](/images/equations/Lune/Inline13.svg) |
(4)
|
where the lengths of the horizontal and vertical legs of the triangle are 
and 
,
respectively.
