circumference

{{For|the circumference of a graph|Circumference (graph theory)}}

[[File:Circle-withsegments.svg|thumb|{{legend-line|black solid 3px|circumference C}}

{{legend-line|green solid 2px|center or origin O}} Circumference = {{pi}} × diameter = 2{{pi}} × radius.]]

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.{{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}} More generally, the perimeter is the curve length around any closed figure.

Circumference may also refer to the circle itself, that is, the locus corresponding to the edge of a disk.

The {{em|{{visible anchor|circumference of a sphere}}}} is the circumference, or length, of any one of its great circles.

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 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 is 1, its circumference is $\pi.$ ]]

File:2pi-unrolled.gif 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 constants. This constant, pi, is represented by the Greek letter $\pi.$ Its first few decimal digits are 3.141592653589793...{{Cite OEIS|A000796}} Pi is defined as the ratio of a circle's circumference $C$ to its diameter $d:$ {{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}}

$\pi = \frac{C}{d}.$

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

${C} = \pi \cdot{d} = 2\pi \cdot{r}.\!$

The ratio of the circle's circumference to its radius is equivalent to $2\pi$ .{{efn|The Greek letter {{tau}} (tau) is sometimes used to represent this constant. This notation is accepted in several online calculators{{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}} and many programming languages.{{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}}{{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}}{{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}}}} This is also the number of radians in one 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 $C/d,$ 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=[https://archive.org/details/historyofmathema00katz/page/109 109]|url-access=registration|url=https://archive.org/details/historyofmathema00katz/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

File:Ellipses same circumference.png

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 ellipse,

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,$

$C_{\rm{ellipse}} \sim \pi \sqrt{2\left(a^2 + b^2\right)}.$

Some lower and upper bounds on the circumference of the canonical ellipse with $a\geq b$ are:{{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 }}

$2\pi b \leq C \leq 2\pi a,$

$\pi (a+b) \leq C \leq 4(a+b),$

$4\sqrt{a^2+b^2} \leq C \leq \pi \sqrt{2\left(a^2+b^2\right)}.$

Here the upper bound $2\pi a$ is the circumference of a circumscribed concentric circle passing through the endpoints of the ellipse's major axis, and the lower bound $4\sqrt{a^2+b^2}$ 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|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}} More precisely,

$C_{\rm{ellipse}} = 4a \int_0^{\pi/2} \sqrt{1 - e^2 \sin^2\theta}\ d\theta,$

where $a$ is the length of the semi-major axis and $e$ is the eccentricity $\sqrt{1 - b^2/a^2}.$

Notes

References

External links

[http://www.numericana.com/answer/ellipse.htm#elliptic Numericana - Circumference of an ellipse]

Category:Geometric measurement

Category:Circles