the area of a partial lune is equal to the area of a kite.

see attached picture:

the out side square is set at sqrt(2)/2 " and the internal squares are set at 1/2" and 90 degree offset

the internal inscribed circle has a diameter = sqrt(2/2)"

all arcs have a radius of 1/2" from the 8 square points.

4 internal quadrant squares are halfed by right angle triangles for a mid line and then the intersecting 90 degree off set line from the other square is mirrored {repeat for all 4 quadrents}

then form the internal middle square from those segments.

from there it is a easy process of identifying the identical parts and manipulating them to prove the kite = the partial lune.

anyone have a better way to write this out mathematically instead of a "picto" gram solution.