Problem:
Four circles of radius are arranged as shown. Their centers are the vertices of a square. The area of the shaded region is closest to
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Answer Choices:
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Solution:
The four quarter-circles that lie inside the square have a total area equal to the area of one of the circles, . The length of a side of the square is equal to two radii, , and thus the square has area . The difference is , so it is closest to . (The area, to one decimal place, is .)
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions