1985 AMC 12 第 2 题

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

剩余圆弧所对的圆心角为

300^\circ

The remaining circular arc has central angle

300^\circ

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

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.

其他年份的第 2 题