Proving Circle Has Smallest Perimeter for Same Area

[pixel01]

Hi everybody,

In physics and mathematics, we often use the theorem that the circle alway has the smallest perimeter compared to all other geometric shapes of the same area. Anyone can tell me how to prove that?
Thanks for reading.

 

[arildno]

Well, assuming the perimeter is a smooth curve, you can show with calculus of variations that the circle minimizes the perimeter for a given area.

I haven't seen the proof valid for ALL perimeters, but I'm sure it exists somewhere.