Problem:
Three circles with radius are mutually tangent. What is the total area of the circles and the region bounded by them, as shown in the figure?
Answer Choices:
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Solution:
Connect the centers of the three circles to form an equilateral triangle with side length . Then the requested area is equal to the area of this triangle and a sector of each circle. The equilateral triangle has base and altitude , so its area is
The area of each sector is . Hence the total area is .
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The problems on this page are the property of the MAA's American Mathematics Competitions