Greatest area using a string with the length of $l$

Question

Suppose we have a string with length of $l$ what is the shape that has highest area?

In other words,with a constant perimeter of $l$ what is the shape with the highest area?

P.S:My own speculation is circle with the perimeter of $l$ but I can't really prove it.

Answer 1

Your intuition is correct. The isoperimetric inequality states that a shape with area $A$ and perimeter $P$ (your $l$) satisfies $$A \le \frac{P^2}{4\pi}.$$ Equality holds in the case of a circle. If the radius is $r$ then $A = \pi r^2 = \dfrac{(2 \pi r)^2}{4\pi} = \dfrac{P^2}{4\pi}$.

Both the proof of the isometric inequality and the fact that equality holds only for circles aren't entirely trivial. The article in the link is a good place to start looking for references.

Answer 2

In Calulus of Variations it can be shown that when all possibilities are considered the maximum area is:

$$ \pi [\frac{l}{2 \pi}]^2 $$.

If not looped around so that the ends are on a straight line the solution is a semi-circle.

Answer 3

If you assume that there is an area-maximizing shape, it's easy to see it has to be a circle, as otherwise a small deformation will produce a shape with greater area.