
Construction of the lune:
 We draw an isosceles rectangular triangle
 The arc of circumference $$AB$$ is created by the circle centred in $$O$$
 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.
 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 $$OA=OB$$.