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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 (A=pir_A^2) with same perimeter (p=2pir_p) as the curve,

Q=(r_A)/(r_p^2)
(1)
=((A/pi))/((p/(2pi))^2)
(2)
=(4piA)/(p^2),
(3)

where A is the area of the plane figure and p is its perimeter. The isoperimetric inequality gives Q<=1, with equality only in the case of the circle.

IsoperimetricQuotient

For a regular n-gon with inradius r, the area is given by

 A=nr^2tan(pi/n),
(4)

edge length by

 a=2rtan(pi/n),
(5)

and the perimeter is given by

 p=na=2nrtan(pi/n).
(6)

Thus,

 Q_n=pi/(ntan(pi/n)),
(7)

which converges to 1 for n->infty.

The isoperimetric quotient can similarly be defined for a polyhedron, where it is defined as the dimensionless quantity obtained using the volume (V=4pir_V^3/3) and surface area (S=4pir_S^2) of the sphere as a reference,

Q=(r_V^2)/(r_S^3)
(8)
=((3/(4pi)V)^2)/((1/(4pi)S)^3)
(9)
=(36piV^2)/(S^3).
(10)

See also

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

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