Question

Calculate the perimeter of this sector. ​

Answer 1

The two straight edges of the shape correspond to the original circle's radius, 5.5 cm. We're told this sector has area 30.25 cm^2, which we will use to determine the measure of the central angle subtended by the arc (the remaining edge of the shape).

The complete circle has an area of $\pi(5.5\,\rm cm)^2$, which corresponds to a "central angle" of $2\pi\,\rm rad$. Then if $\theta$ is the central angle of this sector, we have

$\dfrac{\pi(5.5\,\rm cm)^2}{2\pi\,\rm rad}=\dfrac{30.25\,\mathrm{cm}^2}\theta\implies\theta=2\,\mathrm{rad}$

The complete circle has a circumference of $2\pi(5.5\,\rm cm)$. Then if $\ell$ is the length of the sector's arc, we get

$\dfrac{2\pi(5.5\,\rm cm)}{2\pi\,\rm rad}=\dfrac{\ell}{2\,\rm rad}\implies\ell=11\,\rm cm$

So the sector has a total perimeter of

5.5 cm + 5.5 cm + 11 cm = 22 cm