The lune

1. Construction of the lune:
2. 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

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

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