jmc

Let the two sides of the rectangle be $a$ and $b$. The problem is now telling us $ab=2a+2b$. Putting everything on one side of the equation, we have $ab-2a-2b=0.$ This looks tricky. However, we can add a number to both sides of the equation to make it factor nicely. 4 works here: $$ab-2a-2b+4=4\Rightarrow (a-2)(b-2)=4$$Since we don't have a square, $a$ and $b$ must be different. It doesn't matter which one is which, so we can just say $a=6$ and $b=3$. The perimeter is then $2(6+3)=\boxed{18}$