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# Isoperimetric Quotient

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) (2) (3)

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)

edge length by

 (5)

and the perimeter is given by

 (6)

Thus,

 (7)

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) (9) (10)

Isoperimetric Inequality, Kelvin's Conjecture

Portions of this entry contributed by Hermann Kremer

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## References

Croft, H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 23, 1991.

## Referenced on Wolfram|Alpha

Isoperimetric Quotient

## Cite this as:

Kremer, Hermann and Weisstein, Eric W. "Isoperimetric Quotient." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/IsoperimetricQuotient.html