Assuming the ceiling has a rectangular shape, the perimeter is the sum of the lengths of all the sides, hence twice the sum of length and width.
2(8 1/2 + 10 9/12) = 2(18 +6/12 +9/12)
= 2(18 +15/12)
= 2(19 3/12)
= 2(19 1/4)
= 38 1/2 . . . . . the room's perimeter
