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{{Short description|Perimeter of a circle or ellipse}}
{{For|the circumference of a graph|Circumference (graph theory)}}
[[File:Circle-withsegments.svg|thumb|{{legend-line|black solid 3px|circumference ''C''}}
{{legend-line|blue solid 2px|diameter ''D''}}
{{legend-line|red solid 2px|radius ''R''}}
{{legend-line|green solid 2px|center or origin ''O''}} Circumference = {{pi}} × diameter = 2{{pi}} × radius.]]
{{General geometry}}

In [[geometry]], the '''circumference''' ({{etymology|la|{{wikt-lang|la|circumferēns}}|carrying around, circling}}) is the [[perimeter]] of a [[circle]] or [[ellipse]]. The circumference is the [[arc length]] of the circle, as if it were opened up and straightened out to a [[line segment]].<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> More generally, the perimeter is the [[curve length]] around any closed figure.
Circumference may also refer to the circle itself, that is, the [[Locus (geometry)|locus]] corresponding to the [[Edge (geometry)|edge]] of a [[Disk (geometry)|disk]].
The {{em|{{visible anchor|circumference of a sphere}}}} is the circumference, or length, of any one of its [[great circle]]s.

== Circle ==
{{redirect|2πr|the TV episode|2πR (Person of Interest){{!}}2πR (''Person of Interest'')}}
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 <math>\pi.</math>]]
[[File:2pi-unrolled.gif|thumb|240px|When a circle's [[radius]] is 1—called a [[unit circle]]—its circumference is <math>2\pi.</math>]]

=== 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)|<math>\pi.</math>]] Its first few decimal digits are 3.141592653589793...<ref>{{Cite OEIS|A000796}}</ref> Pi is defined as the [[ratio]] of a circle's circumference <math>C</math> to its [[diameter]] <math>d:</math><ref>{{Cite web |title=Mathematics Essentials Lesson: Circumference of Circles |url=https://openhighschoolcourses.org/mod/book/view?id=258&chapterid=502 |access-date=2024-12-02 |website=openhighschoolcourses.org}}</ref>
<math display="block">\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 display=block>{C} = \pi \cdot{d} = 2\pi \cdot{r}.\!</math>

The ratio of the circle's circumference to its radius is equivalent to <math>2\pi</math>.{{efn|The Greek letter {{tau}} (tau) is sometimes used to represent [[Tau (mathematical constant)|this constant]]. This notation is accepted in several online calculators<ref name="Desmos">{{cite web |title=Supported Functions |url=https://help.desmos.com/hc/en-us/articles/212235786-Supported-Functions |access-date=2024-10-21 |website=help.desmos.com |url-status=live |archive-url=https://web.archive.org/web/20230326032414/https://help.desmos.com/hc/en-us/articles/212235786-Supported-Functions |archive-date=2023-03-26}}</ref> and many programming languages.<ref name="Python_370">{{cite web |title=math — Mathematical functions |work=Python 3.7.0 documentation |url=https://docs.python.org/3/library/math.html#math.tau |access-date=2019-08-05 |url-status=live |archive-url=https://web.archive.org/web/20190729033443/https://docs.python.org/3/library/math.html |archive-date=2019-07-29}}</ref><ref name="Java-docs">{{cite web |title=Math class |website=Java 19 documentation |url=https://docs.oracle.com/en/java/javase/19/docs/api/java.base/java/lang/Math.html#TAU}}</ref><ref name="Rust">{{cite web |title=std::f64::consts::TAU - Rust |url=https://doc.rust-lang.org/stable/std/f64/consts/constant.TAU.html |access-date=2024-10-21 |website=doc.rust-lang.org |url-status=live |archive-url=https://web.archive.org/web/20230718194313/https://doc.rust-lang.org/stable/std/f64/consts/constant.TAU.html |archive-date=2023-07-18}}</ref>}} This is also the number of [[radian]]s in one [[Turn_(angle)|turn]]. 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 (written as <math>C/d,</math> 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=[https://archive.org/details/historyofmathema00katz/page/109 109]|url-access=registration|url=https://archive.org/details/historyofmathema00katz/page/109}}</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 ==
[[File:Ellipses same circumference.png|thumb|Circle, and ellipses with the same circumference]]
{{Main|Ellipse#Circumference}}
Some authors use circumference 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 form|canonical]] ellipse,
<math display=block>\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,</math>
is
<math display=block>C_{\rm{ellipse}} \sim \pi \sqrt{2\left(a^2 + b^2\right)}.</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 |issue=499|year=2014|pages=227–234|doi=10.2307/3621497|jstor=3621497|s2cid=126427943 }}</ref>
<math display=block>2\pi b \leq C \leq 2\pi a,</math>
<math display=block>\pi (a+b) \leq C \leq 4(a+b),</math>
<math display=block>4\sqrt{a^2+b^2} \leq C \leq \pi \sqrt{2\left(a^2+b^2\right)}.</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|s2cid=119810884|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|issue=7|mr=966232|doi=10.2307/2323302|jstor=2323302}}</ref> More precisely,
<math display=block>C_{\rm{ellipse}} = 4a \int_0^{\pi/2} \sqrt{1 - e^2 \sin^2\theta}\ d\theta,</math>
where <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>

== See also ==
* {{annotated link|Arc length}}
* {{annotated link|Area}}
* {{annotated link|Circumgon}}
* {{annotated link|Isoperimetric inequality}}
* {{annotated link|Perimeter-equivalent radius}}

==Notes==
{{Notelist}}

==References==
{{Reflist}}

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

{{Authority control}}

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

Circumference - Revision history

imported>Twinsday: add ety + wikt templates