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 [tex]\pi(5.5\,\rm cm)^2[/tex], which corresponds to a "central angle" of [tex]2\pi\,\rm rad[/tex]. Then if [tex]\theta[/tex] is the central angle of this sector, we have[tex]\dfrac{\pi(5.5\,\rm cm)^2}{2\pi\,\rm rad}=\dfrac{30.25\,\mathrm{cm}^2}\theta\implies\theta=2\,\mathrm{rad}[/tex]The complete circle has a circumference of [tex]2\pi(5.5\,\rm cm)[/tex]. Then if [tex]\ell[/tex] is the length of the sector's arc, we get[tex]\dfrac{2\pi(5.5\,\rm cm)}{2\pi\,\rm rad}=\dfrac{\ell}{2\,\rm rad}\implies\ell=11\,\rm cm[/tex]So the sector has a total perimeter of5.5 cm + 5.5 cm + 11 cm = 22 cm