View source for Circumference

[[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.<ref>{{cite web | url=http://www-rohan.sdsu.edu/~pwbrock/files/UNIT9.3.pdf | title =Perimeter, Area and Circumference | author =[[San Diego State University]] | publisher =[[Addison-Wesley]] | year =2004| archive-url=https://web.archive.org/web/20141006153741/http://www-rohan.sdsu.edu/~pwbrock/files/UNIT9.3.pdf|archive-date=6 October 2014}}</ref> 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 (geometry)|edge (boundary)]] of a [[Disk (geometry)|disk]], circumference is a special case of [[perimeter]].<ref>{{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}} </ref> 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 [[ellipse]]s.
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 (mathematics)|limit]] of the perimeters of inscribed [[regular polygon]]s as the number of sides increases without bound.<ref>{{citation|first=Harold R.|last=Jacobs|title=Geometry|year=1974|publisher=W. H. Freeman and Co.|isbn=0-7167-0456-0|page=565}}</ref> 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 [[diameter]] is 1, its circumference is {{pi}}.]]
[[File:2pi-unrolled.gif|thumb|240px|When a circle's [[radius]] is 1—called a [[unit circle]]—its circumference is 2{{pi}}.]]

=== Relationship with {{pi}} ===
The circumference of a [[circle]] is related to one of the most important [[mathematical constant]]s. This [[Constant (mathematics)|constant]], [[pi]], is represented by the [[Greek letter]] [[Pi (letter)|{{pi}}]]. The first few decimal digits of the numerical value of {{pi}} are 3.141592653589793 ...<ref>{{Cite OEIS|A000796}}</ref> Pi is defined as the [[ratio]] of a circle's circumference {{math|''C''}} to its [[diameter]] {{math|''d''}}:

:<math> \pi = \frac{C}{d}.</math>

Or, equivalently, as the ratio of the circumference to twice the [[radius]]. The above formula can be rearranged to solve for the circumference:

:<math>{C}=\pi\cdot{d}=2\pi\cdot{r}.\!</math>

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.<ref>{{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|url-access=registration|url=https://archive.org/details/historyofmathema00katz}}</ref> 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 10<sup>40</sup> sides.

== Ellipse ==
{{main|Ellipse#Circumference}}
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, 
:<math>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,</math>
is 
:<math>C_{\rm{ellipse}} \sim \pi \sqrt{2(a^2 + b^2)}.</math>
Some lower and upper bounds on the circumference of the canonical ellipse with <math>a\geq b</math> are<ref>{{cite journal|last1=Jameson|first1=G.J.O.|title=Inequalities for the perimeter of an ellipse| journal= Mathematical Gazette|volume= 98 |year=2014|pages=227–234|doi=10.2307/3621497}}</ref>

:<math>2\pi b\le C\le 2\pi a,</math>

:<math>\pi (a+b)\le C\le 4(a+b),</math>

:<math>4\sqrt{a^2+b^2}\le C\le \pi \sqrt{2(a^2+b^2)} .</math>

Here the upper bound <math>2\pi a</math> is the circumference of a [[circumscribed circle|circumscribed]] [[concentric circle]] passing through the endpoints of the ellipse's major axis, and the lower bound <math>4\sqrt{a^2+b^2}</math> is the [[perimeter]] of an [[inscribed figure|inscribed]] [[rhombus]] with [[vertex (geometry)|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]].<ref>{{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}}</ref> More precisely, we have
:<math>C_{\rm{ellipse}} = 4a\int_0^{\pi/2}\sqrt {1 - e^2 \sin^2\theta}\ d\theta,</math>
where again <math>a</math> is the length of the semi-major axis and <math>e</math> is the eccentricity <math>\sqrt{1 - b^2/a^2}.</math>

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

== See also ==
* [[Arc length]]
* [[Area]]
* [[Isoperimetric inequality]]

==References==
{{Reflist}}

== External links ==
{{wikibooks|Geometry|Circles/Arcs|Arcs}}
{{Wiktionary|circumference}}
* [http://www.numericana.com/answer/ellipse.htm#elliptic Numericana - Circumference of an ellipse]

[[Category:Geometric measurement]]
[[Category:Circles]]