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.
A = (pi) (R²)
R = √(64/pi).
Circumference = (2 pi) (R) = 2 pi √(64/pi) = √(256 pi) = 28.359 (rounded).
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 run.
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 maximum 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.
