Ingeometry, thecircumference (from Latincircumferēns'carrying around, circling') is theperimeter of acircle orellipse. The circumference is thearc length of the circle, as if it were opened up and straightened out to aline segment.[1] More generally, the perimeter is thecurve length around any closed figure. Circumference may also refer to the circle itself, that is, thelocus corresponding to theedge of adisk. Thecircumference of a sphere is the circumference, or length, of any one of itsgreat circles.
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 thelimit of the perimeters of inscribedregular polygons as the number of sides increases without bound.[2] The term circumference is used when measuring physical objects, as well as when considering abstract geometric forms.
When a circle'sdiameter is 1, its circumference isWhen a circle'sradius is 1—called aunit circle—its circumference is
Or, equivalently, as the ratio of the circumference to twice theradius. The above formula can be rearranged to solve for the circumference:
The ratio of the circle's circumference to its radius is equivalent to.[a] This is also the number ofradians in oneturn. The use of the mathematical constantπ is ubiquitous in mathematics, engineering, and science.
InMeasurement of a Circle written circa 250 BCE,Archimedes showed that this ratio (written as since he did not use the nameπ) was greater than 310/71 but less than 31/7 by calculating the perimeters of an inscribed and a circumscribed regular polygon of 96 sides.[9] This method for approximatingπ 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 byChristoph Grienberger who used polygons with 1040 sides.
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 thesemi-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 thecanonical ellipse,isSome lower and upper bounds on the circumference of the canonical ellipse with are:[10]
Here the upper bound is the circumference of acircumscribedconcentric circle passing through the endpoints of the ellipse's major axis, and the lower bound is theperimeter of aninscribedrhombus withvertices at 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.[11] More precisely,where is the length of the semi-major axis and is the eccentricity
Perimeter-equivalent radius – Radius of a circle or sphere equivalent to a non-circular or non-spherical objectPages displaying short descriptions of redirect targets
^The Greek letter 𝜏 (tau) is sometimes used to representthis constant. This notation is accepted in several online calculators[5] and many programming languages.[6][7][8]
^Bennett, Jeffrey; Briggs, William (2005),Using and Understanding Mathematics / A Quantitative Reasoning Approach (3rd ed.), Addison-Wesley, p. 580,ISBN978-0-321-22773-7
^Jacobs, Harold R. (1974),Geometry, W. H. Freeman and Co., p. 565,ISBN0-7167-0456-0
^Almkvist, Gert; Berndt, Bruce (1988), "Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses,π, and the Ladies Diary",American Mathematical Monthly,95 (7):585–608,doi:10.2307/2323302,JSTOR2323302,MR0966232,S2CID119810884
