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?

A. $\sqrt{2}$

B. 2

C. $2\sqrt{2}$

D. 4

E. $4\sqrt{2}$

1. 2025 AMC 8 官方解析

Answer (B): The circle on the left has a radius of 1 and an area of $\pi$ square units. Its inscribed square has a diagonal length of 2 and a side length of $\sqrt{2}$, so the area of the square is 2 square units. Thus the shaded region on the left has an area of $\pi - 2$ square units.

The radius of the larger circle on the right is $R$, so the area of the circle is $\pi R^2$ square units. Its inscribed square has a diagonal length of $2R$ and a side length of $\sqrt{2}R$, so the area of the square is $2R^2$ square units. Thus the shaded region on the right has an area of $\frac{1}{4} (\pi R^2 - 2R^2) = \frac{1}{4} R^2(\pi - 2)$ square units.

Setting the two shaded areas equal gives $\frac{1}{4} R^2(\pi - 2) = \pi - 2 \Longrightarrow R^2 = 4 \Longrightarrow R = 2$.

Therefore the radius $R$ of the circle on the right is 2 units.

OR

Let $A$ equal the area of the shaded region on the left. Suppose the smaller circle on the left is enlarged to match the size of the larger circle on the right. Then the radius of the circle on the left will grow by a factor of $R$ and the shaded area on the left will grow by a factor of $R^2$ to produce a new shaded area of $AR^2$. Only one of the four shaded sectors in the new circle is included in the circle on the right. This means the shaded region on the right equals $AR^2 / 4$. The shaded regions in the original figures have the same area, so $AR^2 / 4 = A$, which implies $R^2 / 4 = 1$, thus $R = 2$. Therefore the radius $R$ of the circle on the right is 2 units.