Why does a circle enclose the largest area? In this wikipedia, article http://en.wikipedia.org/wiki/Circle#Area_enclosed its stated that the circle is the closed curve which has the maximum area for a given arc length. First, of all, I would like to see different proofs, for this result. (If there are any elementary ones!)
One, interesting observation, which one can think while seeing this problem, is: How does one propose such type of problem? Does, anyone take all closed curves, and calculate their area to come this conclusion? I don't think that's the right intuition.

May I use an intuitive answer here because it may be easy enough for primary school students to think about.
Let's imagine a wire which is bent into two sides. If we want to "enclose" an area as large as possible, any two adjacent sides must be as far away from each other as possible. (Compare with the "enclosed area" when the two sides are very close to each other.) That means the angle between two adjacent sides must tend to 180 degree. But since the length of the wire is fixed and the area need to be enclosed, each side should be as short as possible in order to get larger adjacent angles.
Eventually the shape will become a circle.
