1985 AMC 12 Problem 2

Question

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

The remaining circular arc has central angle

300^\circ

Add the length of that arc to the two exposed radii

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.

Problem 2 in other years