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Wondering if it is possible to estimate the surface area of a 2D shape where the only known value is the length of the enclosing boundary, and that it is know the internal surface area is solid. Really have no idea if it's possible, though if it is, what would be the most simple formula that would take the enclosing boundary as an input, and output the 2D shape's surface area?

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The maximum area is obtained using a circle. There is no minimum. – André Nicolas Apr 28 '12 at 1:24
+1 @André Nicolas: Makes sense, feel free to post your comment as is, as an answer. Thanks! – blunders Apr 28 '12 at 1:25
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An answer has been given. – André Nicolas Apr 28 '12 at 1:27
+1 @André Nicolas: As you wish, your comment was a huge help, and was noted below the answer. Cheers! – blunders Apr 28 '12 at 1:30
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For polygons with a given number $n$ of sides, the best, as you might expect, is the regular $n$-gon. There are calculus proofs, but I strongly recommend Ivan Niven's book Maxima and Minima Without Calculus. – André Nicolas Apr 28 '12 at 1:51

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It's not possible. You can get triangles of arbitrarily small area with a fixed perimeter. However, as you probably imagine, you cannot have arbitrarily large area with a fixed perimeter: see the isoperimetric inequality.

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+1 @lhf: Thanks, the visuals on the wikipage for "isoperimetric inequality", combined with André Nicolas comment "The maximum area is obtained using a circle. There is no minimum", confirms it is not possible; to me. Big thanks for the amazingly fast answer! Cheers! – blunders Apr 28 '12 at 1:31

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