Why does a circle have the largest area for a given perimeter?

[Amith2006]

Homework Statement

For a two DIMENSIONAL figure of given perimeter why does the circle have the largest area? Similarly, of all three dimensional figures of given volume the sphere has the largest area. why?

Homework Equations

The Attempt at a Solution

 

[lzkelley]

You can set up a differential equation describing the relationship between area and perimeter for varying shapes... solving for the maximum results in the equation for a circle. The same thing in 3D results in a sphere.

 

[Amith2006]

thanx buddy.

 

[HallsofIvy]

The complete answer involves "Calculus of Variations".