Question

下列图形由正方形和圆组成。哪个图形的阴影区域面积最大？

The following figures are composed of squares and circles. Which figure has a shaded region with largest area?

只有 $A$

$A$ only

只有 $B$

$B$ only

只有 $C$

$C$ only

$A$ 和 $B$ 都是

both $A$ and $B$

全部相等

all are equal

在图形 $A$ 中，阴影面积是一个 $2$ 乘 $2$ 正方形面积减去半径为 $1$ 的圆面积，即 $4-\pi$ 。

图形 $B$ 由四个图形 $A$ 的半尺寸副本组成，所以总阴影面积也为 $4-\pi$ 。

在图形 $C$ 中，圆的半径为 $1$ ，内接正方形的对角线为 $2$ ，所以其面积为 $2$ 。阴影面积为 $\pi-2$ 。

因为 $\pi>3$ ，所以 $\pi-2>4-\pi$ ，因此图形 $C$ 的阴影面积最大。

所以正确答案是 C。

In figure $A$ , the shaded area is the area of a $2$ by $2$ square minus a circle of radius $1$ , so it is $4-\pi$ .

Figure $B$ is made of four half-size copies of figure $A$ , so its total shaded area is also $4-\pi$ .

In figure $C$ , the circle has radius $1$ , and the inscribed square has diagonal $2$ , so its area is $2$ . The shaded area is $\pi-2$ .

Since $\pi>3$ , we have $\pi-2>4-\pi$ , so figure $C$ has the largest shaded area.

Thus, C is the correct answer.