If it needs to be a rectangle, then the rectangle with the smallest perimeter for a given area is the square. He needs 32 feet of fence, and should fence off a square that's 8 x 8 .
But if he's willing to go to the trouble, the perimeter of a circle with the same area is even less than the square.
That's 11.4% less fence to buy, for a circular run.
But on the other hand, what have you got against the dog ? One of the two main purposes of a dog run is to give the dog a place to <u>run</u>. Minimizing the perimeter also minimizes the distance where he can get up some speed and run in a straight line ... freeing up his hips, clearing the cobwebs from his brain, smelling the air, keeping his claws nice and worn down. With the emotional well-being of the dog in mind, I'd expect you'd want to give him the <u>maximum</u> possible straight route inside the run, which, unfortunately, also maximizes the amount of fence that Malcolm has to provide.
But I digress. The math is done. The question is answered. This case is closed.
