Challenge Level

A *lune* is the area left when part of a circle is cut off by another circle, as in the following problems. It is called a lune because it looks a bit like the moon.

- In the following figure, two semicircles have been drawn, one on the side $AB$ of the triangle, and the other on the side $AC$ of the triangle (with centre $O$). What is the area of the blue (shaded) lune which is bounded by the two semicircles?

As a bonus, can you construct a square on the diagram with the same area as the blue lune, using only a straight edge (ruler) and compasses? This is called the*quadrature (making into a square) of the lune*.

- In the following figure, three semicircles have been drawn, one on each of the sides of the right-angled $6$-$8$-$10$ triangle. What is the total area of the two coloured (shaded) lunes in the drawing?