Circumference

For the circumference of a graph, seeCircumference (graph theory).

Ingeometry,thecircumference(from Latincircumferens,meaning "carrying around" ) is theperimeterof acircleorellipse.The circumference is thearc lengthof the circle, as if it were opened up and straightened out to aline segment.More generally, the perimeter is thecurve lengtharound any closed figure. Circumference may also refer to the circle itself, that is, thelocuscorresponding to theedgeof adisk. Thecircumference of a sphereis the circumference, or length, of any one of itsgreat circles.

circumferenceC

diameterD

radiusR

center or originO

Circumference =π× diameter = 2π× radius.

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 thelimitof the perimeters of inscribedregular polygonsas the number of sides increases without bound.The term circumference is used when measuring physical objects, as well as when considering abstract geometric forms.

When a circle'sdiameteris 1, its circumference is$\pi.$

When a circle'sradiusis 1—called aunit circle—its circumference is$2\pi.$

Relationship withπ

The circumference of acircleis related to one of the most importantmathematical constants.Thisconstant,pi,is represented by theGreek letter$\pi.$Its first few decimal digits are 3.141592653589793....Pi is defined as theratioof a circle's circumference$C$to itsdiameter$d:$ $$\pi ={\frac {C}{d}}.$$

Or, equivalently, as the ratio of the circumference to twice theradius.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 called the circle constant, and is equivalent to$2\pi$.The value$2\pi$is also the amount ofradiansin oneturn.The use of the mathematical constantπis ubiquitous in mathematics, engineering, and science.

InMeasurement of a Circlewritten circa 250 BCE,Archimedesshowed that this ratio (written as$C/d,$since he did not use the nameπ) was greater than 3⁠10/71⁠but less than 3⁠1/7⁠by calculating the perimeters of an inscribed and a circumscribed regular polygon of 96 sides.This method for approximatingπwas used for centuries, obtaining more accuracy by using polygons of larger and larger number of sides. The last such calculation was performed in 1630 byChristoph Grienbergerwho used polygons with 10⁴⁰sides.

Ellipse

Circle, and ellipses with the same circumference

Main article: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 thesemi-major and semi-minor axesof 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 thecanonicalellipse, $${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1,$$ is $$C_{\rm {ellipse}}\sim \pi {\sqrt {2\left(a^{2}+b^{2}\right)}}.$$ Some lower and upper bounds on the circumference of the canonical ellipse with$a\geq b$are: $$2\pi b\leq C\leq 2\pi a,$$ $$\pi (a+b)\leq C\leq 4(a+b),$$ $$4{\sqrt {a^{2}+b^{2}}}\leq C\leq \pi {\sqrt {2\left(a^{2}+b^{2}\right)}}.$$

Here the upper bound$2\pi a$is the circumference of acircumscribedconcentric circlepassing through the endpoints of the ellipse's major axis, and the lower bound$4{\sqrt {a^{2}+b^{2}}}$is theperimeterof aninscribedrhombuswithverticesat the endpoints of the major and minor axes.

The circumference of an ellipse can be expressed exactly in terms of thecomplete elliptic integral of the second kind.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${\sqrt {1-b^{2}/a^{2}}}.$

See also

• Arc length– Distance along a curve
• Area– Size of a two-dimensional surface
• Circumgon– Geometric figure which circumscribes a circle
• Isoperimetric inequality– Geometric inequality which sets a lower bound on the surface area of a set given its volume
• Perimeter-equivalent radius– Radius of a circle or sphere equivalent to a non-circular or non-spherical objectPages displaying short descriptions of redirect targets

External links

• Numericana - Circumference of an ellipse