What shape has the largest area for a fixed perimeter?

[qspeechc]

Hello everyone.

While I was waiting for my computer program to run, I occupied myself with this little problem. For a fixed perimeter, which regular polygon (or any closed shape in the plane) has the largest area? The answer is the circle (I guess), if we regard it as an infinite-sided polygon. But how does one prove this? Are there any simple proofs? Any intuitive proofs?
Obviously then one could look at closed surfaces in R^3, and see which has the largest volume for a fixed surface area, which should be the sphere.
Any thoughts?

Btw, I got this odd relation while trying to work out this problem:
$$\lim_{n\rightarrow \infty}{\frac{\cot{(\pi /n)}}{4n}} = \pi$$
lol.

 

[arildno]

These are called isoperimetric problems, and are dealt with using calculus of variations.

And yes, you have identified the correct shapes that maximize the area/volume for a given perimeter/surface.

 

[qspeechc]

Thanks, I googled "isoperimteric" and got some sites which gave intuitive proofs, that don't use calculus of variations, which is what I wanted.