2004 AMC 10B Problem 9

A square has sides of length

10,

and a circle centered at one of its vertices has radius

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

10^2 = 100

and the circle has area

\pi(10)^2 = 100\pi.

Since the circle is centered at a vertex of the square, exactly one quarter of the circle, area

25\pi,

lies inside the square. The union has area

100 + 100\pi - 25\pi = 100 + 75\pi.

Thus, the correct answer is B.