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# Quadrature of the Lunes

**This is an Underground Mathematics resource.**

*Underground Mathematics is hosted by Cambridge Mathematics. The project was originally funded by a grant from the UK Department for Education to provide free web-based resources that support the teaching and learning of post-16 mathematics.*

*Visit the site at undergroundmathematics.org to find more resources, which also offer suggestions, solutions and teacher notes to help with their use in the classroom.*
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Age 16 to 18

Challenge Level

- Problem
- Student Solutions

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?

This comes in two parts, with the first being less fiendish than the second. Itâ€™s great for practising both quadratics and laws of indices, and you can get a lot from making sure that you find all the solutions. For a real challenge (requiring a bit more knowledge), you could consider finding the complex solutions.

You're invited to decide whether statements about the number of solutions of a quadratic equation are always, sometimes or never true.

This will encourage you to think about whether all quadratics can be factorised and to develop a better understanding of the effect that changing the coefficients has on the factorised form.