2004 AMC 8 第 25 题
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所有题目均经美国数学协会（MAA）官方合法授权使用。
25.
两个 正方形成直角相交，并且每个正方形都平分另一个正方形被其穿过的两条边，如图所示。圆的直径是两个交点之间的线段。从两个正方形中去掉圆后形成的阴影区域面积是多少？
Two squares intersect at right angles, bisecting their intersecting sides, as shown. The circle’s diameter is the segment between the two points of intersection. What is the area of the shaded region created by removing the circle from the squares?
小提示：
求两个正方形并集覆盖的面积
Find the area covered by the union of the two squares.
大提示：
圆的直径是 重叠正方形的对角线
The circle diameter is the diagonal of the overlap square.
解答：
两个 正方形总面积为 ，但它们的重叠部分是一个 、面积为 的正方形。因此两个正方形并集的面积为 。
圆的直径是这个 重叠正方形的对角线，所以直径为 ，半径为 。
圆面积为 ，所以阴影面积为 。
所以正确答案是 D。
The two squares have total area but their overlap is a square with area Thus the area covered by the union of the two squares is
The circle’s diameter is the diagonal of that overlap square, so the diameter is and the radius is
The circle area is so the shaded area is
Thus, D is the correct answer.
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