Question

左图中的圆半径为 $1$ 个单位。圆和内接正方形之间的区域被涂阴影。右图中的圆里，圆和内接正方形之间区域的四分之一被涂阴影。两圆中的阴影区域面积相同。右图圆的半径 $R$ 是多少？

The circle shown below on the left has a radius of $1$ unit. The region between the circle and the inscribed square is shaded. In the circle shown on the right, one quarter of the region between the circle and the inscribed square is shaded. The shaded regions in the two circles have the same area. What is the radius $R,$ in units, of the circle on the right?

$\sqrt{2}$

$2$

$2\sqrt{2}$

$4$

$4\sqrt{2}$

答案：B

知识点：相似长度、面积与体积的缩放关系

难度评级：1310

小提示：

用缩放来比较阴影环形区域面积

Match the shaded gap areas by thinking about scaling

大提示：

面积按长度缩放因子的平方变化

Areas scale as the square of the length scale factor

视频讲解：

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文字解答：

右图中，圆与内接正方形之间的区域只有四分之一被涂上阴影。要使这块阴影面积等于左图中整块区域的面积，右图中圆与正方形之间的整块区域面积必须是左图对应区域的 $4$ 倍。

两个由圆和内接正方形组成的图形是相似的，它们的面积按半径的平方缩放。因此 $R^2 = 4 \cdot 1^2\text{，}$ 所以 $R = 2$ ，即选项 B。

Only one quarter of the gap between the circle and square is shaded on the right. For that shaded area to equal the whole gap on the left, the entire gap on the right must have $4$ times the area of the gap on the left.

The two circle-and-square figures are similar, and their areas scale as the square of their radii. Thus $R^2 = 4 \cdot 1^2,$ so $R = 2,$ which is choice B.

2025 AMC 8 第 18 题