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Polygon of Largest Area.

  1. Nov 3, 2009 #1
    Hello everyone.

    While I was waiting for my computer program to run, I occupied myself with this little problem. For a fixed perimeter, which regular polygon (or any closed shape in the plane) has the largest area? The answer is the circle (I guess), if we regard it as an infinite-sided polygon. But how does one prove this? Are there any simple proofs? Any intuitive proofs?
    Obviously then one could look at closed surfaces in R^3, and see which has the largest volume for a fixed surface area, which should be the sphere.
    Any thoughts?

    Btw, I got this odd relation while trying to work out this problem:
    [tex]\lim_{n\rightarrow \infty}{\frac{\cot{(\pi /n)}}{4n}} = \pi[/tex]
    Last edited: Nov 3, 2009
  2. jcsd
  3. Nov 3, 2009 #2


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    These are called isoperimetric problems, and are dealt with using calculus of variations.

    And yes, you have identified the correct shapes that maximize the area/volume for a given perimeter/surface.
  4. Nov 3, 2009 #3
    Thanks, I googled "isoperimteric" and got some sites which gave intuitive proofs, that don't use calculus of variations, which is what I wanted.
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