In my previous question it was showed to me that given a perimeter, a square would maximize the area. So a square would be more ideal than any other rectangle. My question is what is the most efficient shape such that given a required outside measurement (permitter, circumference... ) it would take up the maximum area. Also how would we prove that such shape is the most efficient.
This is known as the Didochrone (from the myth of Dido) problem, and the solution is the circle. This is more or less the first exercise in most books on variational calculus, so I won't repeat the proof here -- you can easily look it up now that I've given you a pointer.
I should add that I learned the name "Didochrone" from some calculus of variations text, but cannot find it online. I surely didn't invent it. From another website, the origin of the term is fairly clear, though:
"Dido and her followers fled from Tyre, landing on the shores of North Africa. There a local ruler named Iarbas agreed to sell Dido as much land as the hide of a bull could cover. Dido cut a bull's hide into thin strips and used it to outline a large area of land. On that site, Dido built Carthage and became its queen."
Although "isoperimetric" is probably a more precise or direct word for the problem, I personally like "Didochrone"'s classical reference.
There are several different methods for solving this kind of problem (the calculus of variations is perhaps the best known, but other simpler methods are sometimes possible). Although a circle is best when there are no other constraints, there are many interesting variants. For example if you wanted to make a rectangular fence against a long wall, so the constraint is only on the length of three of the sides, the answer would no longer be a square.
You can show this by setting up an equation for the area in terms of one of the sides and the total length of the fence and plotting it.