## Sixteen Circles

#### Problem

Sixteen unit circles are placed together as shown in the diagram.

Find the radius of the shaded circle in the centre.

#### Solution

We begin be drawing a large square and a quarter square on the diagram.

Let us consider the top right quadrant.

Using the Pythagorean Theorem, 2^{2} = `a`^{2} + `a`^{2} `a` = 2.

Similarly, 2^{2} = `b`^{2} + 1^{2} `b` = 3.

As `r` + 2 = 2 + 3, we get `r` = 2 + 3 – 2.

If the centres of the circles in the outside ring of the original diagram are joined, will it form a regular dodecagon?

Problem ID: 186 (01 Nov 2004) Difficulty: 3 Star