Among all closed curves of a given perimeter in the plane, the circle encloses the greatest area, with equality only for the circle. Known in special cases since antiquity, its full rigorous proof came much later; it generalizes to surfaces enclosing volume in higher dimensions.
Facts
StatementFor a closed curve in the plane with perimeter length L enclosing an area A, the isoperimetric inequality states that L squared is always at least 4 times pi times A, with equality holding if and only if the curve is a circle. 1 Classification
Statement Form Statement FormCharacterization Theorem 1 Connections
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Sources
1. Isoperimetric Inequality (Wikipedia)
Wikimedia Foundationlead paragraph, plane isoperimetric inequality statementQuote, lead paragraph, plane isoperimetric inequality statement
and that equality holds if and only if the curve is a circle.
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