# 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".

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

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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