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2004 AMC 12B Problem 7

Problem 7 of 25EasierGeometry

A square has sides of length 10,10, and a circle centered at one of its vertices has radius 10.10. What is the area of the union of the regions enclosed by the square and the circle?

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Solution

The square has area 100100 and the circle has area 100π.100\pi. Their overlap is the quarter of the circle lying inside the square, with area 25π.25\pi. The union is 100+100π−25π=100+75π.100 + 100\pi - 25\pi = 100 + 75\pi. Thus, the correct answer is B.

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Concepts: circle area · area decomposition

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.