Question

如图，一个直径为 $1$ 的圆被从 $2 \times 3$ 的长方形中去掉。阴影区域的面积最接近哪个整数？

A circle of diameter $1$ is removed from a $2 \times 3$ rectangle, as shown. Which whole number is closest to the area of the shaded region?

$1$

$2$

$3$

$4$

$5$

长方形面积是 $2 \times 3 = 6$ ；再减去圆的面积。

The rectangle has area $2 \times 3 = 6$ ; subtract the area of the circle

圆半径为 $\dfrac12$ ，面积为 $\pi\left(\dfrac12\right)^2 = \dfrac{\pi}{4}$ ，略小于 $1$ 。

The circle has radius $\dfrac12,$ so its area is $\pi\left(\dfrac12\right)^2 = \dfrac{\pi}{4},$ a little less than $1$

长方形面积为 $6$ ，去掉的圆面积为 $\pi\left(\dfrac12\right)^2 = \dfrac{\pi}{4} \approx 0.79$ 。

阴影区域面积约为 $6 - 0.79 \approx 5.2$ ，最接近的整数是 $5$ 。

所以正确答案是 E。

The rectangle has area $6,$ and the removed circle has area $\pi\left(\dfrac12\right)^2 = \dfrac{\pi}{4} \approx 0.79.$

So the shaded region has area $6 - 0.79 \approx 5.2,$ whose closest whole number is $5.$

Thus, the correct answer is E .