University of South Carolina High School Math Contest/1993 Exam/Problem 3

Problem

If 3 circles of radius 1 are mutually tangent as shown, what is the area of the gap they enclose?

$\mathrm{(A) \ }\sqrt{3}-\frac{\pi}2 \qquad \mathrm{(B) \ } \frac 16 \qquad \mathrm{(C) \ }\frac 13 \qquad \mathrm{(D) \ } \frac{\sqrt{3}}2 - \frac{\pi}6 \qquad \mathrm{(E) \ } \frac{\pi}6$

Solution

Construct an equilateral triangle whose sidelengths (length $2$ ) are composed of the radii of the circles. The area is thus $\sqrt{3}-\pi/2$ . This problem needs a solution. If you have a solution for it, please help us out by adding it.

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University of South Carolina High School Math Contest/1993 Exam/Problem 3

Problem

Solution