mathschallenge.net

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, 2² = a² + a² a = 2.

Similarly, 2² = b² + 1² 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?