Food for thought

Quadrature of the lunes Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Problem

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.

1. 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.

2. 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?

3. As a final (hard) challenge, can you find any other lunes associated with a circle which can be “squared”; that is, where it is possible to construct a square with the same area as the lune, using only the original circles, a straight edge and compasses? (For example, the lune in question 1 can be squared.)