circumference

American Heritage Dictionary:

cir·cum·fer·ence

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(sər-kŭm'fər-əns) pronunciation

The boundary line of a circle.
1. The boundary line of a figure, area, or object.
2. (Abbr. c or circ.) The length of such a boundary.

[Middle English, from Old French circonference, from Latin circumferentia, from circumferēns, circumferent-, present participle of circumferre, to carry around : circum-, circum- + ferre, to carry.]

circumferential cir·cum'fer·en'tial (-fə-rĕn'shəl) adj.

SYNONYMS circumference, circuit, compass, perimeter, periphery. These nouns refer to a line around a closed figure or area: the circumference of the earth; followed the circuit around the park; stayed within the compass of the schoolyard; the perimeter of a rectangle; a fence around the periphery of the property.

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noun

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Antonyms by Answers.com:

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Definition: edge, perimeter
Antonyms: inside, interior, middle

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To convert from circumference to:

radians, multiply by 6.283.

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IN BRIEF: The line that bounds a rounded figure.

The circumference of a circle is the exact same distance from the middle of the circle all the way around.

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Dictionary of Cultural Literacy: Science:

circumference

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(suhr-kum-fuhr-uhns)

The measure of the distance around a circle.

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For a list of words related to circumference, see:

Quantities, Relationships, and Operations - circumference: distance measured around outside of circle
Dimensions and Directions - circumference: outer boundary, esp. perimeter of circle

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Circumference

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This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (June 2010)

When a circle's radius is 1 unit, its circumference is 2

π

units

Circumference =

π

× diameter

mathematical constant $π$
Part of a series of articles on the

Uses
Area of disk Circumference Use in other formulae
Properties
Irrationality Transcendence
Value
Less than 22/7 Approximations Memorization
People
Archimedes Liu Hui Zu Chongzhi Madhava of Sangamagrama William Jones John Machin John Wrench Ludolph van Ceulen Aryabhata
History
Chronology Book
In culture
Legislation Holiday
Related topics
Squaring the circle Basel problem Feynman point Other topics related to $π$
v t e

The circumference is the distance around a closed curve. Circumference is a special perimeter.

Contents

1 Circumference of a circle
2 Circumference of an ellipse
- 2.1 Muir-1883
- 2.2 Ramanujan-1914 (#1,#2)
3 Circumference of a graph
4 External links

Circumference of a circle

The circumference of a circle is the length around it. The circumference of a circle can be calculated from its diameter using the formula:

$c=\pi\cdot{d}.\,\!$

Or, substituting the radius for the diameter:

$c=\pi\cdot{d}=\pi\cdot{2r},\,\!$

where $r$ is the radius and $d$ is the diameter of the circle, and the Greek letter $π$ is defined as the ratio of the circumference of the circle to its diameter. The numerical value of $π$ is 3.141 592 653 589 793....

Circumference of an ellipse

The circumference of an ellipse is more problematic, as the exact solution requires finding the complete elliptic integral of the second kind. This can be achieved either via numerical integration (the best type being Gaussian quadrature) or by one of many binomial series expansions.

Where $a$ , $b$ are the ellipse's semi-major and semi-minor axes, respectively, and $α$ is the ellipse's angular eccentricity,

$\alpha=\arccos\!\left(\frac{b}{a}\right)=2\arctan\!\left(\!\sqrt{\frac{a-b}{a+b}}\,\right);\,\!$

$\begin{align}\mbox{E2}\left[0,90^\circ\right]&= \mbox{Integral}'s\mbox{ divided difference};\\ Pr&=a\times\mbox{E2}\left[0,90^\circ\right] \quad(\mbox{perimetric radius});\\ c&=2\pi\times Pr.\end{align}\,\!$

There are many different approximations for the $E2 [0,90°]$ divided difference, with varying degrees of sophistication and corresponding accuracy.

In comparing the different approximations, the $\tan^2\!\left(\frac{\alpha}{2}\right)\,\!$ (also known as " $n$ ", the third flattening of the ellipse) based series expansion is used to find the actual value:

$\begin{align}\mbox{E2}\left[0,90^\circ\right] &=\cos^2\!\left(\frac{\alpha}{2}\right)\frac{1}{UT}\sum_{TN=1}^{UT=\infty}{.5\choose{}TN}^2\tan^{4TN}\!\left(\frac{\alpha}{2}\right),\\ &=\cos^2\!\left(\frac{\alpha}{2}\right)\Bigg(1+\frac{1}{4}\tan^4\!\left(\frac{\alpha}{2}\right) +\frac{1}{64}\tan^8\!\left(\frac{\alpha}{2}\right)\\ &\qquad\qquad\qquad\;\,+\frac{1}{256}\tan^{12}\!\left(\frac{\alpha}{2}\right) +\frac{25}{16384}\tan^{16}\!\left(\frac{\alpha}{2}\right) +...\Bigg);\end{align}\,\!$

More generally, the arc length of a portion of the circumference, as a function of the angle subtended, is given by an incomplete elliptic integral.

The inverse function, the angle subtended as a function of the arc length, is given by the elliptic functions.

Muir-1883

Probably the most accurate to its given simplicity is Thomas Muir's:

$\begin{align}Pr &\approx\left(\frac{a^{1.5}+b^{1.5}}{2}\right)^\frac{1}{1.5}=a\left(\frac{2+\cos^{3}\!\left(\alpha\right)}{2}\right)^\frac{2}{3},\\ &\quad\approx{a}\times\cos^2\!\left(\frac{\alpha}{2}\right)\left(1+\frac{1}{4}\tan^4\!\left(\frac{\alpha}{2}\right)\right).\end{align}\,\!$

Ramanujan-1914 (#1,#2)

Srinivasa Ramanujan introduced two different approximations, both from 1914.

$\begin{align}1.\;Pr&\approx\frac{1}{2}\Big(3(a+b)-\sqrt{\big(3a+b\big)\big(a+3b\big)}\Big),\\ &\quad=\frac{a}{2}\bigg(6\cos^2\!\left(\frac{\alpha}{2}\right)-\sqrt{\big(3+\cos\!\left(\alpha\right)\big)\big(1+3\cos\!\left(\alpha\right)\big)}\bigg);\end{align}\,\!$

$\begin{align}2.\;Pr&\approx\frac{1}{2}\Big(a+b\Big)\Bigg(1+\frac{3\big(\frac{a-b}{a+b}\big)^2}{10+\sqrt{4-3\big(\frac{a-b}{a+b}\big)^2}}\Bigg);\\ &\quad=a\times\cos^2\!\left(\frac{\alpha}{2}\right)\Bigg(1+\frac{3\tan^4\!\big(\frac{\alpha}{2}\big)}{10+\sqrt{4-3\tan^4\!\big(\frac{\alpha}{2}\big)}}\Bigg).\end{align}\,\!$

The second equation is demonstrably by far the better of the two, and is among the most accurate approximations known.

Letting $a = 10000$ and $b = a cos{oε}$ , results with different ellipticities can be found and compared:

b	Pr	Ramanujan-#2	Ramanujan-#1	Muir
9975	9987.50391 11393	9987.50391 11393	9987.50391 11393	9987.50391 11389
9966	9983.00723 73047	9983.00723 73047	9983.00723 73047	9983.00723 73034
9950	9975.01566 41666	9975.01566 41666	9975.01566 41666	9975.01566 41604
9900	9950.06281 41695	9950.06281 41695	9950.06281 41695	9950.06281 40704
9000	9506.58008 71725	9506.58008 71725	9506.58008 67774	9506.57894 84209
8000	9027.79927 77219	9027.79927 77219	9027.79924 43886	9027.77786 62561
7500	8794.70009 24247	8794.70009 24240	8794.69994 52888	8794.64324 65132
6667	8417.02535 37669	8417.02535 37460	8417.02428 62059	8416.81780 56370
5000	7709.82212 59502	7709.82212 24348	7709.80054 22510	7708.38853 77837
3333	7090.18347 61693	7090.18324 21686	7089.94281 35586	7083.80287 96714
2500	6826.49114 72168	6826.48944 11189	6825.75998 22882	6814.20222 31205
1000	6468.01579 36089	6467.94103 84016	6462.57005 00576	6431.72229 28418
100	6367.94576 97209	6366.42397 74408	6346.16560 81001	6303.80428 66621
10	6366.22253 29150	6363.81341 42880	6340.31989 06242	6299.73805 61141
1	6366.19804 50617	6363.65301 06191	6339.80266 34498	6299.60944 92105
iota	6366.19772 36758	6363.63636 36364	6339.74596 21556	6299.60524 94744

Circumference of a graph

In graph theory the circumference of a graph refers to the longest cycle contained in that graph.

External links

Look up circumference in Wiktionary, the free dictionary.

Numericana - Circumference of an ellipse
Circumference of a circle With interactive applet and animation

This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)

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Translations:

Circumference

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Dansk (Danish)
n. - omkreds, periferi

Nederlands (Dutch)
cirkelomtrek

Français (French)
n. - circonférence

Deutsch (German)
n. - Umfang, Kreislinie

Ελληνική (Greek)
n. - περιφέρεια ή περίμετρος κύκλου

Italiano (Italian)
circonferenza

Português (Portuguese)
n. - circunferência (f), âmbito (m), limite (m)

Русский (Russian)
окружность

Español (Spanish)
n. - circunferencia

Svenska (Swedish)
n. - omkrets, periferi

中文（简体）(Chinese (Simplified))
圆周, 胸围, 周围

中文（繁體）(Chinese (Traditional))
n. - 圓周, 胸圍, 周圍

한국어 (Korean)
n. - 원주, 주위, 영역

日本語 (Japanese)
n. - 円周, 周辺, 周囲

العربيه (Arabic)
‏(الاسم) محيط الدائرة, محيط‏

עברית (Hebrew)
n. - ‮היקף‬

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American Heritage Dictionary The American Heritage® Dictionary of the English Language, Fourth Edition Copyright © 2007, 2000 by Houghton Mifflin Company. Updated in 2009. Published by Houghton Mifflin Company. All rights reserved. Read

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Dictionary of Cultural Literacy: Science The New Dictionary of Cultural Literacy, Third Edition Edited by E.D. Hirsch, Jr., Joseph F. Kett, and James Trefil. Copyright © 2002 by Houghton Mifflin Company. Published by Houghton Mifflin. All rights reserved. Read

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Wikipedia on Answers.com This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article Circumference. Read