Question

If the perimiter of a sector is 4 times its radius, then the radian measure of the ceenral angel of the sector is

Answer 1

The perimeter of a sector includes two segments that are each equal to the radius, and the arc length. For your sector, the arc length is found as
   perimeter = 4r = r + r + arc-length
   2r = arc-length

Now we also know that arc-length is related to the central angle (in radians) by
   arc-length = r*central-angle
   2r = r*central-angle
   2 = central-angle

The measure of the central angle of the sector is 2 radians.