Circumference
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In geometry, the circumference (from Latin circumferentia, meaning "carrying around") of a circle is the (linear) distance around it.[1] That is, the circumference would be the length of the circle if it were opened up and straightened out to a line segment. Since a circle is the edge (boundary) of a disk, circumference is a special case of perimeter.[2] The perimeter is the length around any closed figure and is the term used for most figures excepting the circle and some circular-like figures such as ellipses.
Informally, "circumference" may also refer to the edge itself rather than to the length of the edge.
Contents
1 Circumference of a circle
1.1 Relationship with π
2 Circumference of an ellipse
3 Circumference of a graph
4 See also
5 References
6 External links
Circumference of a circle
Circle illustration with circumference (C) in black, diameter (D) in cyan, radius (R) in red, and centre or origin (O) in magenta. Circumference = π × diameter = 2 × π × radius.
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 can not 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.[3] The term circumference is used when measuring physical objects, as well as when considering abstract geometric forms.
When a circle's diameter is 1, its circumference is π.
When a circle's radius is 1—called a unit circle—its circumference is 2π.
Relationship with π
The circumference of a circle is related to one of the most important mathematical constants. This constant, pi, is represented by the Greek letter π. The first few decimal digits of the numerical value of π are 3.141592653589793 ...[4] Pi is defined as the ratio of a circle's circumference C to its diameter d:
- π=Cd.displaystyle pi =frac Cd.
Or, equivalently, as the ratio of the circumference to twice the radius. The above formula can be rearranged to solve for the circumference:
- C=π⋅d=2π⋅r.displaystyle C=pi cdot d=2pi cdot r.!
The use of the mathematical constant π is ubiquitous in mathematics, engineering, and science.
In Measurement of a Circle written circa 250 BCE, Archimedes showed that this ratio (C/d, since he did not use the name π) 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.[5] This method for approximating π 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.
Circumference of an ellipse
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 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,
- x2a2+y2b2=1,displaystyle frac x^2a^2+frac y^2b^2=1,
is
- Cellipse∼π2(a2+b2).displaystyle C_rm ellipsesim pi sqrt 2(a^2+b^2).
Some lower and upper bounds on the circumference of the canonical ellipse with a≥bdisplaystyle ageq b
- 2πb≤C≤2πa,displaystyle 2pi bleq Cleq 2pi a,
- π(a+b)≤C≤4(a+b),displaystyle pi (a+b)leq Cleq 4(a+b),
- 4a2+b2≤C≤π2(a2+b2).displaystyle 4sqrt a^2+b^2leq Cleq pi sqrt 2(a^2+b^2).
Here the upper bound 2πadisplaystyle 2pi a
The circumference of an ellipse can be expressed exactly in terms of the complete elliptic integral of the second kind.[7] More precisely, we have
- Cellipse=4a∫0π/21−e2sin2θ dθ,displaystyle C_rm ellipse=4aint _0^pi /2sqrt 1-e^2sin ^2theta dtheta ,
where again adisplaystyle a
Circumference of a graph
In graph theory the circumference of a graph refers to the longest (simple) cycle contained in that graph.[8]
See also
- Arc length
- Area
- Isoperimetric inequality
References
^ San Diego State University (2004). "Perimeter, Area and Circumference" (PDF). Addison-Wesley. Archived from the original (PDF) on 6 October 2014.
^ Bennett, Jeffrey; Briggs, William (2005), Using and Understanding Mathematics / A Quantitative Reasoning Approach (3rd ed.), Addison-Wesley, p. 580, ISBN 978-0-321-22773-7
^ Jacobs, Harold R. (1974), Geometry, W. H. Freeman and Co., p. 565, ISBN 0-7167-0456-0
^ Sloane, N.J.A. (ed.). "Sequence A000796". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
^ Katz, Victor J. (1998), A History of Mathematics / An Introduction (2nd ed.), Addison-Wesley Longman, p. 109, ISBN 978-0-321-01618-8
^ Jameson, G.J.O. (2014). "Inequalities for the perimeter of an ellipse". Mathematical Gazette. 98: 227–234. doi:10.2307/3621497.
^ Almkvist, Gert; Berndt, Bruce (1988), "Gauss, Landen, Ramanujan, the arithmetic-geometric mean, ellipses, π, and the Ladies Diary", American Mathematical Monthly, 95: 585–608, doi:10.2307/2323302, MR 0966232
^ Harary, Frank (1969), Graph Theory, Addison-Wesley, p. 13, ISBN 0-201-02787-9
External links
 | The Wikibook Geometry has a page on the topic of: Arcs |
 | Look up circumference in Wiktionary, the free dictionary. |
- Numericana - Circumference of an ellipse
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