The area of the circle circumscribing three identical circles touching each other is \frac{\pi(2+\sqrt{3})^2}{3} square cm. What is the radius of one of the smaller circles ?

Explanation

Let the radius of the smaller circles be r. Their centers form an equilateral triangle of side 2r. The distance from the centroid to a vertex is \frac{2r}{\sqrt{3}}. The circumradius R = \frac{2r}{\sqrt{3}} + r = r(\frac{2+\sqrt{3}}{\sqrt{3}}). The area is \pi R^2 = \pi r^2 \frac{(2+\sqrt{3})^2}{3}. Equating this to the given area, r^2 = 1 \implies r = 1 cm.

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