Which shape optimize the perimeter for a given area?

Question

I was wondering what is shape that maximize the perimeter for a given area?

In fact I would like to know what would be the most optimized perimeter that I can include in a rectangle. See image below.

Thanks!

Answer 1

Suppose you enclose an area equal to $1/3$ of the area of the rectangle. Divide one edge of the rectangle into $n$ equal segments. On each small segment construct an isosceles triangle with height $2/3$ of the width of the rectangle. The total areas of these triangles is $n \times \frac{2/3 \times \ell \times L/n}{2} = 1/3 \ell \times L$.

The sum of the perimeters of these triangles is larger than the sum of the heights which is $n\times \ell \times 2/3$. This quantity is unbounded as $n$ increases. Therefore, there is no configuration which maximizes the perimeter given the enclosed area.