How can you prove Mathematically that the maximum area enclosed by a given length is a circle?
The precise statement of the theorem:
Of all plane figures with a given perimeter, the circle has the greatest area.
Alternatively, the theorem can be stated, Of all plane figures with a given area, the circle has the least perimeter.
Elementary proofs can be found in many books under the heading "the isoperimetric theorem." It is sometimes also referred to as "Dido's problem" inspired by the legend of the founding of Carthage. Here are two appropriate references:
Nicholas D. Kazarinoff, GEOMETRIC INEQUALITIES. (New Mathematical Library #4 -- it is now published by the Mathematical Association of America). See chapter 2.
Ivan Niven, MAXIMA AND MINIMA WITHOUT CALCULUS. (DOLCIANI MATHEMATICAL EXPOSITIONS NO. 6. Published by the Mathematical Association of America). See section 4.3 starting on page 81.
There are many easy proofs to be found, but beware that there is a subtle point involved, which is not relevant to many of us; but as a consequence, you can also find quite sophisticated proofs.Cheers,