Problem Solving With Areas

When finding the area of a composite shape or part of a composite shape, this is often symetry that can be used. The diagram below shows three overlapping circles, each of radiusThe centres of the circles are at A, B and C respectively and each circle passes through the centre of each of the other circles. The intersection of all three circles is shaded and the problem is to find this area.

Because A is at the centre of a circle, radiusand this circle also passes through B, the distanceBy symmetry,is then an equolateral triangle with side of length and area

Cut the shaded area up into an equilateral triangle and three equal segments 1, 2, 3 as shown below.

The area of each segment is

The shaded area is the area of the triangle plus the area of the three segments.