2004 AMC 10B Problem 9

Question

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?

200 + 25\pi

100 + 75\pi

75 + 100\pi

100 + 100\pi

100 + 125\pi

Add the areas of the square and the circle, then subtract the overlap

The overlap is the quarter of the circle that lies inside the square

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.

Problem 9 in other years