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.