how do you find the length of the perimeter of a shaded area in a sector

Assuming we know the radius (or diameter) and the area of the sector....we have

Area of sector  = (1/2)(r^2)Θ   multiply both sides by 2 and divide by r^2

2*Area of sector/(r^2) =  Θ       

Then using the value you obtain for Θ , we have

Arc length of sector = r*Θ