I'm tyring to understand how the isoperimetric inequality came into existence. It seems like finding the region which yields maximum area when enclosed by a curve of fixed length is an old problem. Queen Dido seems to have figured out the solution a long time ago: that the region should be a circle.
I'm currently reading how Steiner's method of characterizing how certain regions cannot yield maximum area.
My question however is that how was the following inequality formulated:
$4\pi A\leq L^2$
where $A$ is the area of the region which we will be considering and $L$ is the length of the curve which encloses it.
I understand we are trying to find a relationship between the area of the region and the length of the curve. I expect the area to be a function of the length perhaps ($A$~$f(L)$ where ~ represents some relation). In this case it happens that we have $A \leq \frac{1}{4\pi}L^2$
My question is simply -- why?
EDIT
Given a region $\mathbb{L}$ with $A=Area(\mathbb{L})$ and a curve with a fixed length of $L$
why do we have that $4\pi A\leq L^2$ ?