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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?"

Silent
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1 Answers1

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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.

Hagen von Eitzen
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