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circumference
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(File:Circle-withsegments.svg|thumb|Circumference (C in black) of a circle with diameter (D in cyan), radius (R in red), and centre (O in magenta). Circumference = {{pi}} × diameter = 2 × {{pi}} × radius.)In geometry, the circumference (from Latin circumferens, meaning "carrying around") of a circle is the (linear) distance around it.WEB,weblink Perimeter, Area and Circumference, San Diego State University, Addison-Wesley, 2004,weblink" title="web.archive.org/web/20141006153741weblink">weblink 6 October 2014, That is, the circumference would be the length of the circle if it were opened up and straightened out to a line segment. Since a circle is the edge (boundary) of a disk, circumference is a special case of perimeter.{{citation|first1=Jeffrey|last1=Bennett|first2=William|last2=Briggs|title=Using and Understanding Mathematics / A Quantitative Reasoning Approach|edition=3rd|publisher=Addison-Wesley|year=2005|isbn=978-0-321-22773-7|page=580}} The perimeter is the length around any closed figure and is the term used for most figures excepting the circle and some circular-like figures such as ellipses.Informally, "circumference" may also refer to the edge itself rather than to the length of the edge.

Circle

The circumference of a circle is the distance around it, but if, as in many elementary treatments, distance is defined in terms of straight lines, this cannot be used as a definition. Under these circumstances, the circumference of a circle may be defined as the limit of the perimeters of inscribed regular polygons as the number of sides increases without bound.{{citation|first=Harold R.|last=Jacobs|title=Geometry|year=1974|publisher=W. H. Freeman and Co.|isbn=0-7167-0456-0|page=565}} The term circumference is used when measuring physical objects, as well as when considering abstract geometric forms.File:Pi-unrolled-720.gif|thumb|240px|When a circle's diameterdiameterFile:2pi-unrolled.gif|thumb|240px|When a circle's radius is 1—called a unit circleunit circle

Relationship with {{pi}}

The circumference of a circle is related to one of the most important mathematical constants. This constant, pi, is represented by the Greek letter {{pi}}. The first few decimal digits of the numerical value of {{pi}} are 3.141592653589793 ...OEIS, A000796, Pi is defined as the ratio of a circle's circumference {{math|C}} to its diameter {{math|d}}:
pi = frac{C}{d}.
Or, equivalently, as the ratio of the circumference to twice the radius. The above formula can be rearranged to solve for the circumference:
{C}=picdot{d}=2picdot{r}.!
The use of the mathematical constant {{pi}} is ubiquitous in mathematics, engineering, and science. In Measurement of a Circle written circa 250 BCE, Archimedes showed that this ratio ({{math|C/d}}, since he did not use the name {{pi}}) was greater than 3{{sfrac|10|71}} but less than 3{{sfrac|1|7}} by calculating the perimeters of an inscribed and a circumscribed regular polygon of 96 sides.{{citation|first=Victor J.|last=Katz|title=A History of Mathematics / An Introduction|edition=2nd|year=1998|publisher=Addison-Wesley Longman|isbn=978-0-321-01618-8|page=109}} This method for approximating {{pi}} was used for centuries, obtaining more accuracy by using polygons of larger and larger number of sides. The last such calculation was performed in 1630 by Christoph Grienberger who used polygons with 1040 sides.

Ellipse

Circumference is used by some authors to denote the perimeter of an ellipse. There is no general formula for the circumference of an ellipse in terms of the semi-major and semi-minor axes of the ellipse that uses only elementary functions. However, there are approximate formulas in terms of these parameters. One such approximation, due to Euler (1773), for the canonical ellipse,
frac{x^2}{a^2} + frac{y^2}{b^2} = 1,
is
C_{rm{ellipse}} sim pi sqrt{2(a^2 + b^2)}.
Some lower and upper bounds on the circumference of the canonical ellipse with ageq b areJOURNAL, Jameson, G.J.O., Inequalities for the perimeter of an ellipse, Mathematical Gazette, 98, 2014, 227–234, 10.2307/3621497,
2pi ble Cle 2pi a,
pi (a+b)le Cle 4(a+b),
4sqrt{a^2+b^2}le Cle pi sqrt{2(a^2+b^2)} .
Here the upper bound 2pi a is the circumference of a circumscribed concentric circle passing through the endpoints of the ellipse's major axis, and the lower bound 4sqrt{a^2+b^2} is the perimeter of an inscribed rhombus with vertices at the endpoints of the major and minor axes.The circumference of an ellipse can be expressed exactly in terms of the complete elliptic integral of the second kind.{{citation|first1=Gert|last1=Almkvist|first2=Bruce|last2=Berndt|title=Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, {{pi}}, and the Ladies Diary|journal=American Mathematical Monthly|year=1988|pages=585–608|volume=95|mr=966232|doi=10.2307/2323302}} More precisely, we have
C_{rm{ellipse}} = 4aint_0^{pi/2}sqrt {1 - e^2 sin^2theta} dtheta,
where again a is the length of the semi-major axis and e is the eccentricity sqrt{1 - b^2/a^2}.

Graph

In graph theory the circumference of a graph refers to the longest (simple) cycle contained in that graph.{{citation|first=Frank|last=Harary|title=Graph Theory|year=1969|publisher=Addison-Wesley|isbn=0-201-02787-9|page=13}}

See also

References

{{Reflist}}

External links

{{Wiktionary|circumference}}

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