Something in Common: Two crescents and a triangle

Circle crescents introduction

The ‘Something in Common’ collection from John Burke contains problems that draw together different areas of mathematics. The materials are ideal to use in enhancing problem solving skills and in teaching for understanding.  

The ‘Two Crescents and a Triangle’ resource is an extension of the ‘Four Crescents’ resource featured previously. It draws together work on area of a circle, composite shapes, Pythagoras’ theorem and algebra.

Three semi-circular arcs are drawn so that their diameters form a right-angled triangle.

Constructing the crescents

The lengths of two of the diameters are given. The task is to work out what area of the diagram is shaded. Here is an example to try: 

Four crescents problem

There are 16 examples of the problem, and in each case it turns out that the total shaded area is equal to twice the area of the triangle. Why is this so?

Looking at the general case, with lengths a and b given on the triangle, we begin by calculating the square of the hypotenuse of the triangle.

Calculating the hypotenuse

The next step is to calculate the area of the semi-circle whose diameter forms the hypotenuse of the triangle.

Area of semi-cicle

Considering the semi-circles on the legs of the triangle gives:

Calculating shaded area

The next step is to use the areas already calculated to work out the area of the crescents.

Area of crescents

Adding the area of the triangle to the area of the crescents gives the total shaded area.

Calculating total shaded area

Hence, the area of the crescents is equal to the area of the triangle, and the total area is double the area of the triangle. When looking at numerical examples, the final answer is simply the product of the two given lengths.

We have recently added the complete collection of resources from John Burke’s ‘Something in Common’ materials, which comprise; investigations, puzzles and other enrichment activities.

You can also join us at the STEM Centre to experience ideas for teaching through understanding on Enthuse funded courses, details of which are here.

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Subject(s)Mathematics
Age11-14, 14-16
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