Three circles of radii 7.1, 10.9, and 13.5 cm are mutually tangent (see below). Find the shaded area bounded by the three circles. HINT: Connect the centers of the 3 circles.

Question

Three circles of radii 7.1, 10.9, and 13.5 cm are mutually tangent (see below). Find the shaded area bounded by the three circles. HINT: Connect the centers of the 3 circles.

Answer 1

There is a fair amount of brute computation involved to solve this, but I think you do it as follows.

Remember that the line of centers passes through the point of tangency,

You first need to compute the area of the triangle formed by connecting the line of centers.

Since you have the side lengths you can use Heron's formula).

Then you need to solve the triangle for the 3 vertex angles of the triangle.

You can then compute the area of the 3 sectors involved by first computing the area of each circle and then the proportion of that area occupied by the sector.

Then subtract the sum of the areas of the sectors from the area of the triangle.

Happy calculating!