Editing Circumference (section)

== 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.