2024 AMC 8 第 18 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。
18.
以 为圆心的三个同心圆半径分别为 、 和 。点 和 在最大圆上。两个较小圆之间的区域被涂阴影，两个较大圆之间由圆心角 截出的部分也被涂阴影，如下图所示。若阴影区域和非阴影区域面积相等， 的度数是多少？
Three concentric circles centered at have radii of and Points and lie on the largest circle. The region between the two smaller circles is shaded, as is the portion of the region between the two larger circles bounded by central angle as shown in the figure below. Suppose the shaded and unshaded regions are equal in area. What is the measure of in degrees?
小提示：
先算半径 和 之间阴影圆环的面积
Compute the area of the shaded annulus between radii and
大提示：
令阴影面积等于非阴影面积，解圆心角
Set shaded area equal to unshaded area and solve for the sector angle
解答：
设 为 的度数。
阴影区域的一部分是半径 圆与半径 圆之间的圆环，面积为 。另一部分是最大圆的一个扇形减去半径 的圆，面积为 。所以阴影面积为 。
非阴影区域由最小圆和外圆环未阴影部分组成，面积为 。
令两部分面积相等并解 ：
所以正确答案是 A。
Let be the measure of
One component of the shaded region is the area of the circle with radius minus the area of the circle with radius This part has area The remaining area is a sector of the biggest circle minus the area of the circle with radius . This has area Hence, the total area of the shaded region is
Next, we note that the unshaded region is composed of the smallest circle and the unshaded portion of the outer ring. This will have a total area of
Lastly, we equate the area of both regions and solve for
Thus, A is the correct answer.
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