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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
Circumference
of a circle

1.1 Relationship with π

2 Circumference
Circumference
of an ellipse 3 Circumference
Circumference
of a graph 4 See also 5 References 6 External links

Circumference
Circumference
of a circle[edit]

Circle
Circle
illustration with circumference (C) in black, diameter (D) in cyan, radius (R) in red, and centre or origin (O) in magenta. Circumference
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[edit] 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 =

C d

.

displaystyle 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

= π ⋅

d

= 2 π ⋅

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
Circle
written circa 250 BCE, Archimedes
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
Circumference
of an ellipse[edit] Main article: Ellipse
Ellipse
§ Circumference 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 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,

x

2

a

2

+

y

2

b

2

= 1 ,

displaystyle frac x^ 2 a^ 2 + frac y^ 2 b^ 2 =1,

is

C

e l l i p s e

∼ π

2 (

a

2

+

b

2

)

.

displaystyle C_ rm ellipse sim pi sqrt 2(a^ 2 +b^ 2 ) .

Some lower and upper bounds on the circumference of the canonical ellipse with

a ≥ b

displaystyle ageq b

are[6]

2 π b ≤ C ≤ 2 π a ,

displaystyle 2pi bleq Cleq 2pi a,

π ( a + b ) ≤ C ≤ 4 ( a + b ) ,

displaystyle pi (a+b)leq Cleq 4(a+b),

4

a

2

+

b

2

≤ C ≤ π

2 (

a

2

+

b

2

)

.

displaystyle 4 sqrt a^ 2 +b^ 2 leq Cleq pi sqrt 2(a^ 2 +b^ 2 ) .

Here the upper bound

2 π a

displaystyle 2pi a

is the circumference of a circumscribed concentric circle passing through the endpoints of the ellipse's major axis, and the lower bound

4

a

2

+

b

2

displaystyle 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.[7] More precisely, we have

C

e l l i p s e

= 4 a

0

π

/

2

1 −

e

2

sin

2

⁡ θ

  d θ ,

displaystyle C_ rm ellipse =4aint _ 0 ^ pi /2 sqrt 1-e^ 2 sin ^ 2 theta dtheta ,

where again

a

displaystyle a

is the length of the semi-major axis and

e

displaystyle e

is the eccentricity

1 −

b

2

/

a

2

.

displaystyle sqrt 1-b^ 2 /a^ 2 .

Circumference
Circumference
of a graph[edit] In graph theory the circumference of a graph refers to the longest (simple) cycle contained in that graph.[8] See also[edit]

Arc length Area Isoperimetric inequality

References[edit]

^ San Diego State University
San Diego State University
(2004). "Perimeter, Area
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[edit]

The Wikibook Geometry
Geometry
has a page on the topic of: Arcs

Look up circumference in Wiktionary, the free dictionary.

Numericana - Circumference
Circumference
o

.