# What is a direct proof of isoperimetric inequality?

What is a direct proof of isoperimetric inequality? In another word, i am asking for a proof that the circle has the maximum area compare to other geometric shape with the same circumstance but without the assumption that the circle is the "optimal choice".

Could anybody answer?

Comment 1: i think the circle S1 absolutly comes from nowhere, then immediatialy substitude some type of inequality...

Here is the elemetary indirect proof from Do Carmo's Differential Geometry:

Page 1 of the indirect proof: http://i.stack.imgur.com/UXDnD.png

-
What do you mean by "the assumption that the circle is the 'optimal choice'"? That's generally the conclusion of the isoperimetric inequality, not its hypothesis. – Owen Biesel Oct 12 '12 at 20:58
@OwenBiesel - i think the circle S1 absolutly comes from nowhere, then immediatialy substitude some type of inequality... – Victor Oct 12 '12 at 21:20
What do you mean by "$S^1$ absolutely comes from nowhere"? – Neal Oct 12 '12 at 21:39
@neal - I assumed that no mathematician define the circle so far, how could one discover this inequality? – Victor Oct 12 '12 at 21:55
I'm still not sure what you mean. You can't conclude that the circle is optimal without defining the circle. – Neal Oct 13 '12 at 1:00