Question

在一款街机游戏中，“怪物”是图中半径为 $1$ 厘米的圆的阴影扇形。缺失部分（嘴巴）的圆心角为 $60^\circ$ 。怪物的周长是多少厘米？

In an arcade game, the “monster” is the shaded sector of a circle of radius $1$ cm, as shown in the figure. The missing piece (the mouth) has central angle $60^\circ.$ What is the perimeter of the monster in cm?

$\pi+2$

$2\pi$

$\frac53\pi$

$\frac56\pi+2$

$\frac53\pi+2$

答案：E

知识点：弧圆圆周长周长

难度评级：1000

小提示：

剩余圆弧所对的圆心角为 $300^\circ$

The remaining circular arc has central angle $300^\circ$

大提示：

将该圆弧的长度与两条露出的半径相加

Add the length of that arc to the two exposed radii

解答：

曲线部分是单位圆周长的 $\frac{300^\circ}{360^\circ}=\frac{5}{6}$ ，所以其长度为 $(\frac{5}{6})(2\pi)=\frac{5\pi}{3}$ 。嘴巴的两条直边都是半径，总长为 $2$ 。因此周长为 $\frac{5\pi}{3}+2$ 。

所以正确答案是 E。

The curved part is $\frac{300^\circ}{360^\circ}=\frac{5}{6}$ of a unit circle, so its length is $(\frac{5}{6})(2\pi)=\frac{5\pi}{3}.$ The two straight sides of the mouth are radii of total length $2.$ Hence the perimeter is $\frac{5\pi}{3}+2.$

Thus the correct answer is E.

1985 AMC 12 第 2 题

答案：E

难度评级：1000

小提示：

大提示：

解答：

其他年份的第 2 题