jmc

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: $$ab-6a-6b+36=36⟹(a-6)(b-6)=36$$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.