What is the solution to Dido's problem?

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The discussion centers on Dido's problem, which seeks the geometric shape that maximizes area given a fixed perimeter with one side as a straight line. While sources suggest a semi-circle is the optimal solution, calculations show that a full circle provides a larger area when not constrained by a straight edge. However, since Dido's problem specifies a straight side, a semi-circle cannot be the answer. The conclusion is that for Dido's problem, the optimal shape is not a circle but rather a semi-circle due to the straight edge requirement. Understanding the constraints is crucial for determining the correct geometric solution.
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Given a fixed perimeter, what is the geometric shape (non-self-intersecting closed curve) that would maximize the enclosed area? MathWorld (I'm not allowed to link) and a few other sites insist that the answer is a semi-circle, but simple math with perimeter=100 m would show that a circle would have area 2500/\pi\approx795.8 m^{2} whereas a semicircle wold have area \approx594.2 m^{2}.

What am I missing?
 
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Dido's problem demands that one side of the enclosure is a straight line of fixed length. A circle would not qualifiy, since it has no straight sides.

If you allow all closed curves, then the circle would be optimal.
 

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