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 $60^\circ$. 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

First of all, the sum of the lengths of the two radii that make up the mouth is $1+1=2$. There are $360^\circ$ in a circle, so this figure has a circumference $\frac{360^\circ-60^\circ}{360^\circ}=\frac{5}{6}$ of a full circle. A full circle with radius $1$ has a circumference of $2(1)(\pi)=2\pi$, so this has a circumference of $\left(\frac{5}{6}\right)(2\pi)=\frac{5}{3}\pi$. Therefore, the total perimeter is $\frac{5}{3}\pi+2, \boxed{\text{E}}$.

See Also

1985 AHSME (ProblemsAnswer KeyResources)
Preceded by
Problem 1
Followed by
Problem 3
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