The isoperimetric quotient of a closed curve is defined as the ratio of the curve area to the area of a circle () with same perimeter () as the curve,
(1)
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(2)
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where is the area of the plane figure and is its perimeter. The isoperimetric inequality gives , with equality only in the case of the circle.
For a regular -gon with inradius , the area is given by
(4)
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edge length by
(5)
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and the perimeter is given by
(6)
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Thus,
(7)
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which converges to 1 for .
The isoperimetric quotient can similarly be defined for a polyhedron, where it is defined as the dimensionless quantity obtained using the volume () and surface area () of the sphere as a reference,
(8)
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(9)
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(10)
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