If the sides are lengths a and b, the perimeter is 2(a+b) and area=ab=24. b=24/a and the perimeter is 2(a+24/a).
If a and b are integers then the possible values can only be (1,24), (2,12), (3,8), (4,6). The corresponding perimeter values are: 50, 28, 22 and 20. The smallest is 20 (so the sides are 4 and 6).