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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, 22 = a2 + a2 implies a = radical2.

Similarly, 22 = b2 + 12 implies b = radical3.

As r + 2 = radical2 + radical3, we get r = radical2 + radical3 – 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

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