# 1985 AHSME Problems/Problem 2

## Problem

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 $\usepackage{gensymb} 60\degree$. What is the perimeter of the monster in cm?

$[asy] size(100); defaultpen(linewidth(0.7)); filldraw(Arc(origin,1,30,330)--dir(330)--origin--dir(30)--cycle, yellow, black); label("1", (sqrt(3)/4, 1/4), NW); label("60^\circ", (1,0));[/asy]$

$\mathrm{(A)\ } \pi+2 \qquad \mathrm{(B) \ }2\pi \qquad \mathrm{(C) \ } \frac{5}{3}\pi \qquad \mathrm{(D) \ } \frac{5}{6}\pi+2 \qquad \mathrm{(E) \ }\frac{5}{3}\pi+2$

## Solution

The two straight sides of the "monster" are radii of length $1$ cm, so their total length is $1+1 = 2$ cm. The circular arc forms $\frac{360^{\circ}-60^{\circ}}{360^{\circ}} = \frac{5}{6}$ of the entire circle, and a circle with radius $1$ cm has circumference $2\pi(1) = 2\pi$ cm, so the length of the arc is $\left(\frac{5}{6}\right)(2\pi) = \frac{5}{3}\pi$ cm. Hence the total perimeter in cm is $\boxed{\text{(E)} \ \frac{5}{3}\pi+2}$.