In plane geometry, a lune (from Latin luna, meaning "moon") is a shape formed by two curved lines that are parts of circles. One edge of the lune curves outward, and the other curves inward. If you connect two close points on the outward edge, the line goes outside the shape. On the inward edge, the line stays inside the shape. A shape with both edges curving outward is called a lens.

A lune is the area that is inside one circle but not inside another, where the circles overlap but neither is completely inside the other. If there are two circles, A and B, then the part of A that does not overlap with B forms a lune.

Squaring the lune

In the 5th century BC, Hippocrates of Chios proved that the Lune of Hippocrates and two other lunes could be made into a square with the same area using only a straightedge and compass. About the year 1000, Alhazen tried to square a circle using two lunes now named after him. In 1766, the Finnish mathematician Daniel Wijnquist, citing Daniel Bernoulli, identified all five lunes that could be squared. In 1771, Leonhard Euler provided a general method and created an equation related to the problem. In 1933 and 1947, Nikolai Chebotaryov and his student Anatoly Dorodnov proved that only these five lunes can be squared.

Area

The area of a lune created by two circles with radii a and b (where b is larger than a) and a distance c between their centers is calculated using a specific formula. In this formula, cos⁻¹ represents the inverse cosine, also known as arccosine. Additionally, the formula includes the area of a triangle that has sides measuring a, b, and c.