Question

四个半径为 $3$ 的圆这样排列：它们的圆心是一个正方形的四个顶点，并且每个圆都与相邻两个圆相切。阴影区域是该正方形内、所有四个圆之外的部分。阴影区域的面积最接近

Four circles of radius $3$ are arranged so that their centers are the vertices of a square and each circle is tangent to its two neighbors. The shaded region is the part of the square that lies outside all four circles. The area of the shaded region is closest to

$7.7$

$12.1$

$17.2$

$18$

$27$

答案：A

知识点：圆面积面积分割估算

难度评级：1170

解答：

相邻圆相切，所以圆心组成的正方形边长为 $2 \times 3 = 6$ ，面积为 $36$ 。

在正方形内，每个圆贡献一个四分之一圆，四个四分之一圆合成一个完整圆，面积为 $9\pi \approx 28.3$ 。阴影面积为 $36 - 9\pi \approx 7.7$ 。

所以正确答案是 A。

Because adjacent circles are tangent, the square through the centers has side $2 \times 3 = 6$ and area $36.$

Inside the square, each circle contributes a quarter-circle, and the four quarters make one full circle of area $9\pi \approx 28.3.$ The shaded region is $36 - 9\pi \approx 7.7.$

Thus, the correct answer is A .

1992 AMC 8 第 24 题

答案：A

难度评级：1170

解答：

其他年份的第 24 题