A circular field encloses maximal area for minimal perimeter Suppose a farmer has a certain length of fence, $P$ and wished to enclose the largest possible area. What shape area should the farmer choose? Answer is "circle".But, how is it derived? MY TRY: My book says assume that farmer had to enclose rectangular area, and then proceed. So, I used Lagrangian and supposed that $x$ is length and $y$ is breadth and got that $x=y=\frac{P}{4}$, i.e. the area should be square. But, what if the constraint "rectangular area" is omitted? How to show that "a circular field encloses maximal area for minimal perimeter?"

You can approach thge problem by first switching from rectangular to polygonal fields (with fixed side lengths $a$). By a compactness argument, a maximal such polygon exists. If $A,B,C,D$ are four (out of $n$) consecutive points in such a constellation, it is still easy to show that the contribution of $ABCD$ to the total area (under the constraint $AB=BC=CD=a$ and with $A,D$ fixed) is maximal when $BC\|AD$ and hence $\angle CBA = \angle DCB$. This shows that the regular $n$gon is the optimal $n$gon.