Limiting Area of Polygon with given Perimeter

Theorem

Let $\PP$ be the set of plane geometric figures with perimeter $L$.

The element of $P$ with the largest area is the circle of radius $\dfrac L {2 \pi}$ which has area $\dfrac {L^2} {4 \pi}$.

Proof

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Historical Note

This result was demonstrated by Pappus of Alexandria.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $6$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $6$