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January 31st, 2015, 12:29 AM   #1
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Perimeter and area of the circle

How to prove that a circle has the maximum possible area for a given perimeter, and the minimum possible perimeter for a given area.
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January 31st, 2015, 04:17 AM   #2
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This is the so called isoperimetric problem.

Although the circle appears to be an obvious solution to the problem, proving this fact is rather difficult. The first progress toward the solution was made by Swiss geometer Jakob Steiner in 1838, using a geometric method later named Steiner symmetrisation. Steiner showed that if a solution existed, then it must be the circle. Steiner's proof was completed later by several other mathematicians.

Steiner begins with some geometric constructions which are easily understood; for example, it can be shown that any closed curve enclosing a region that is not fully convex can be modified to enclose more area, by "flipping" the concave areas so that they become convex. It can further be shown that any closed curve which is not fully symmetrical can be "tilted" so that it encloses more area. The one shape that is perfectly convex and symmetrical is the circle, although this, in itself, does not represent a rigorous proof of the isoperimetric theorem.
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January 31st, 2015, 05:06 AM   #3
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A general proof would require "Calculus of Variations"- Calculus of Variations -- from Wolfram MathWorld. Are you familiar with that?
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January 31st, 2015, 05:39 AM   #4
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