Dictionary Definition
circumference
Noun
1 the size of something as given by the distance
around it [syn: perimeter]
2 the length of the closed curve of a
circle
User Contributed Dictionary
Synonyms
 (distance measured around any object): girth
Translations
line that bounds a circle or other
twodimensional object
 Finnish: kehä
 Greek: περιφέρεια
 Italian: circonferenza
 Japanese: 円周
 Norwegian: omkrets
 Portuguese: circunferência
 Russian: (for a circle) окружность ; периметр
 Swedish: omkrets
length of such line
 Czech: obvod
 Dutch: omtrek
 Finnish: ympärysmitta
 French: circonférence
 German: Umfang
 Greek: περιφέρεια
 Hebrew:
 Icelandic: ummál
 Italian: circonferenza
 Japanese: 円周長
 Norwegian: omkrets
 Portuguese: comprimento da circunferência, perímetro
 Russian: длина окружности, периметр
 Slovene: obseg
 Spanish: circunferencia
 Swedish: omkrets
Extensive Definition
The circumference is the distance around a
closed
curve. Circumference is a kind of perimeter.
Circumference of a circle
The circumference of a circle can be calculated from its diameter using the formula: c=\pi\cdot.\,\!
Or, substituting the diameter for the radius:
 c=2\pi\cdot=\pi\cdot,\,\!
where r is the radius and d is the diameter of
the circle, and π (the Greek letter pi) is defined as the ratio of the
circumference of the circle to its diameter (the numerical value of
pi is 3.141 592 653 589 793...).
If desired, the above circumference formula can
be derived without reference to the definition of π by using some
integral calculus, as follows:
The upper half of a circle centered at the origin
is the graph of the function f(x) = \sqrt, where x runs from r to
+r. The circumference (c) of the entire circle can be represented
as twice the sum of the lengths of the infinitesimal arcs that make
up this half circle. The length of a single infinitesimal part of
the arc can be calculated using the Pythagorean
formula for the length of the hypotenuse of a rectangular
triangle with side lengths dx and f'(x)dx, which gives us \sqrt =
\left( \sqrt \right) dx.
Thus the circle circumference can be calculated
as
c = 2 \int_^r \sqrtdx = 2 \int_^r \sqrtdx = 2
\int_^r \sqrtdx
The antiderivative needed to
solve this definite integral is the arcsine function:
c = 2r \big[ arcsin(\frac) \big]_^ = 2r \big[
arcsin(1)arcsin(1) \big] = 2r(\tfrac(\tfrac)) = 2\pi r.
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 semimajor
and semiminor
axes, respectively, and o\!\varepsilon\,\! is the ellipse's
angular
eccentricity,
o\!\varepsilon=\arccos\!\left(\frac\right)=2\arctan\!\left(\!\sqrt\,\right);\,\!
\begin\mbox\left[0,90^\circ\right]&=
\mbox's\mbox;\\ Pr&=a\times\mbox\left[0,90^\circ\right]
\quad(\mbox);\\ c&=2\pi\times Pr.\end\,\!
There are many different approximations for the
\mbox\left[0,90^\circ\right] divided
difference, with varying degrees of sophistication and
corresponding accuracy.
In comparing the different approximations, the
\tan\!\left(\frac\right)^2\,\! based series expansion is used to
find the actual value:
\begin\mbox\left[0,90^\circ\right]
&=\cos\!\left(\frac\right)^2
\frac\sum_^^2\tan\!\left(\frac\right)^,\\
&=\cos\!\left(\frac\right)^2\Bigg(1+\frac\tan\!\left(\frac\right)^4
+\frac\tan\!\left(\frac\right)^8\\
&\qquad\qquad\qquad\;\,+\frac\tan\!\left(\frac\right)^
+\frac\tan\!\left(\frac\right)^ +...\Bigg);\end\,\!
Muir1883
 Probably the most accurate to its given simplicity is Thomas Muir's:

 \beginPr
Ramanujan1914 (#1,#2)
 Srinivasa Ramanujan introduced two different approximations, both from 1914

 \begin1.\;Pr&\approx\pi\Big(3(a+b)\sqrt\Big),\\

 \begin2.\;Pr&\approx\frac\Big(a+b\Big)\Bigg(1+\frac\Bigg);\\
 The second equation is demonstratively by far the better of the two, and may be the most accurate approximation known.
Letting a = 10000 and b = a×cos, results with
different ellipticities can be found and compared:
External links
 Numericana  Circumference of an ellipse
 Circumference of a circle With interactive applet and animation
circumference in Bulgarian: Обиколка
(геометрия)
circumference in Catalan: Circumferència
circumference in German: Umfang
circumference in Esperanto: Perimetro
circumference in Estonian: Ümbghfermõõt
circumference in French: Circonférence
circumference in Croatian: Opseg
circumference in Macedonian: Обиколка
circumference in Norwegian Nynorsk:
Omkrins
circumference in Polish: Obwód (geometria)
circumference in Serbian: Обим
(геометрија)
circumference in Finnish: Piiri
(geometria)
circumference in Thai: เส้นรอบวง
Synonyms, Antonyms and Related Words
O, ambit, annular muscle, annulus, areola, aureole, border, boundaries, boundary, bounds, bourns, chaplet, circle, circuit, circumscription,
circus, closed circle,
compass, confines, coordinates, corona, coronet, cortex, covering, crown, crust, cycle, diadem, discus, disk, edges, envelope, epidermis, eternal return,
exterior, external, facade, face, facet, fairy ring, fringe, fringes, front, garland, glory, halo, integument, lasso, limitations, limits, lineaments, logical circle,
loop, looplet, magic circle, marches, margin, metes, metes and bounds, noose, orbit, outer face, outer layer,
outer side, outer skin, outline, outlines, outside, outskirts, pale, parameters, perimeter, periphery, radius, rim, rind, ring, rondelle, round, roundel, saucer, shell, skin, skirts, sphincter, superficies, superstratum, surface, top, verges, vicious circle, wheel, wreath