In an arcade game, the "monster" is the shaded sector of a [circle](https://artofproblemsolving.com/wiki/index.php/Circle) of [radius](https://artofproblemsolving.com/wiki/index.php/Radius) $1$ cm, as shown in the figure. The missing piece (the mouth) has central [angle](https://artofproblemsolving.com/wiki/index.php/Angle) $60^\circ$. What is the [perimeter](https://artofproblemsolving.com/wiki/index.php/Perimeter) of the monster in cm?

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