The lune

1. Construction of the lune:

   1. We draw an isosceles rectangular triangle

   

   2. The arc of circumference $AB$ is created by the circle centred in $O$

   

   3. We draw another arc that passes through $AB$, this time with the center in the middle point of the hypotenuse $AB$:

   

   The blue section is the lune.

2. Area of the lune:

Area of the lune = Area of the semicircle - Area of the circular segment

Area of the lune = Area of the semicircle - Area of the sector + Area of the triangle

$$A_{lune}=\frac{\pi \cdot r^2}{2}-\frac{\pi \cdot r^{\prime 2}\cdot {90}^{\circ }}{{360}^{\circ }}+A_{triangle}$$

where the first $r$ is half of the hypotenuse of the triangle, and the second $r$ is the measure $OA=OB$.