Let the two sides of the rectangle be $ a $ and $ b $. The problem is now telling us $ ab=6a+6b $. Putting everything on one side of the equation, we have, $ ab - 6a - 6b =0 $. This looks tricky. However, we can add a number to both sides of the equation to make it factor nicely. 36 works here: $$ab - 6a - 6b + 36 = 36 \implies (a-6)(b-6)=36$$Since we don't have a square, $ a $ and $ b $ must be different. Thus, the possible factor pairs of $ 36 $ are $ (1,36),(2,18),(3,12),(4,9) $. As we can quickly see, $ 4 + 9 = 13 $ is the smallest sum for any of those pairs, so $ a = 10, b = 15 $, with a total perimeter of $ \boxed{50} $, is the smallest possible perimeter.