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The
circumference is the distance around a closed curve. Circumference is a kind of perimeter.
× diameter
Circle
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=2\pi\cdot{r}=\pi\cdot{2r},\,\!
where
r is the radius and
d is the diameter of the circle, and π (the Greek letter pi) is the
mathematical constant 3.141 592 653 589 793...
== 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 axis and
semi-minor axis axes, respectively, and o\!\varepsilon\,\! is the ellipse's
angular eccentricity,
o\!\varepsilon=\arccos\!\left(\frac{b}{a}\right)=2\arctan\!\left(\!\sqrt{\frac{a-b}{a+b-->\,\right);\,\!
\begin{align}\mbox{E2}\left&= \mbox{Integral}'s\mbox{ divided difference};\\Pr&=a\times\mbox{E2}\left \quad(\mbox{perimetric radius});\\c&=2\pi\times Pr.\end{align}\,\!
There are many different
approximations for the \mbox{E2}\left
Difference quotient, with varying degrees of sophistication and corresponding accuracy.
In comparing the different approximations, the \tan\!\left(\frac{o\!\varepsilon}{2}\right)^2\,\! based series expansion is used to find the actual value:
\begin{align}\mbox{E2}\left&=\cos\!\left(\frac{o\!\varepsilon}{2}\right)^2 \frac{1}{UT}\sum_{TN=1}^{UT=\infty}{.5\choose{}TN}^2\tan\!\left(\frac{o\!\varepsilon}{2}\right)^{4TN},\\&=\cos\!\left(\frac{o\!\varepsilon}{2}\right)^2\Bigg(1+\frac{1}{4}\tan\!\left(\frac{o\!\varepsilon}{2}\right)^4+\frac{1}{64}\tan\!\left(\frac{o\!\varepsilon}{2}\right)^8\\ &\qquad\qquad\qquad\;\,+\frac{1}{256}\tan\!\left(\frac{o\!\varepsilon}{2}\right)^{12}+\frac{25}{16384}\tan\!\left(\frac{o\!\varepsilon}{2}\right)^{16}+...\Bigg);\end{align}\,\!
Muir-1883
Probably the most accurate to its given simplicity is Thomas Muir (mathematician):
:\begin{align}Pr
&\approx\left(\frac{a^{1.5}+b^{1.5-->{2}\right)^\frac{1}{1.5}=a\left(\frac{1+\cos\!\left(o\!\varepsilon\right)^{1.5-->{2}\right)^\frac{1}{1.5},\\&\quad\approx{a}\times\cos\!\left(\frac{o\!\varepsilon}{2}\right)^2\left(1+\frac{1}{4}\tan\!\left(\frac{o\!\varepsilon}{2}\right)^4\right);\end{align}\,\!
Ramanujan-1914 (#1,#2)
Srinivasa Ramanujan introduced
two different approximations, both from 1914
:\begin{align}1.\;Pr&\approx\pi\Big(3(a+b)-\sqrt{\big(3a+b\big)\big(a+3b\big)}\Big),\\
&\quad=\pi{a}\bigg(6\cos\!\left(\frac{o\!\varepsilon}{2}\right)^2\sqrt{\big(3+\cos\!\left(o\!\varepsilon\right)\big)\big(1+3\cos\!\left(o\!\varepsilon\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\!\left(\frac{o\!\varepsilon}{2}\right)^2\Bigg(1+\frac{3\tan\!\big(\frac{o\!\varepsilon}{2}\big)^4}{10+\sqrt{4-3\tan\!\big(\frac{o\!\varepsilon}{2}\big)^4-->\Bigg);\end{align}\,\!
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{
oε}, results with different ellipticities can be found and compared:
{|class="wikitable"|-! 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|}
External links
- Numericana - Circumference of an ellipse
- Circumference of a circle With interactive applet and animation
The
circumference is the distance around a
closed curve. Circumference is a kind of perimeter.
× diameter
Circle
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=2\pi\cdot{r}=\pi\cdot{2r},\,\!
where
r is the radius and
d is the diameter of the circle, and π (the Greek letter pi) is the
mathematical constant 3.141 592 653 589 793...
== 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 axis and semi-minor axis axes, respectively, and o\!\varepsilon\,\! is the ellipse's angular eccentricity,
o\!\varepsilon=\arccos\!\left(\frac{b}{a}\right)=2\arctan\!\left(\!\sqrt{\frac{a-b}{a+b-->\,\right);\,\!
\begin{align}\mbox{E2}\left&= \mbox{Integral}'s\mbox{ divided difference};\\Pr&=a\times\mbox{E2}\left \quad(\mbox{perimetric radius});\\c&=2\pi\times Pr.\end{align}\,\!
There are many different
approximations for the \mbox{E2}\left Difference quotient, with varying degrees of sophistication and corresponding accuracy.
In comparing the different approximations, the \tan\!\left(\frac{o\!\varepsilon}{2}\right)^2\,\! based series expansion is used to find the actual value:
\begin{align}\mbox{E2}\left&=\cos\!\left(\frac{o\!\varepsilon}{2}\right)^2 \frac{1}{UT}\sum_{TN=1}^{UT=\infty}{.5\choose{}TN}^2\tan\!\left(\frac{o\!\varepsilon}{2}\right)^{4TN},\\&=\cos\!\left(\frac{o\!\varepsilon}{2}\right)^2\Bigg(1+\frac{1}{4}\tan\!\left(\frac{o\!\varepsilon}{2}\right)^4+\frac{1}{64}\tan\!\left(\frac{o\!\varepsilon}{2}\right)^8\\ &\qquad\qquad\qquad\;\,+\frac{1}{256}\tan\!\left(\frac{o\!\varepsilon}{2}\right)^{12}+\frac{25}{16384}\tan\!\left(\frac{o\!\varepsilon}{2}\right)^{16}+...\Bigg);\end{align}\,\!
Muir-1883
Probably the most accurate to its given simplicity is
Thomas Muir (mathematician):
:\begin{align}Pr
&\approx\left(\frac{a^{1.5}+b^{1.5-->{2}\right)^\frac{1}{1.5}=a\left(\frac{1+\cos\!\left(o\!\varepsilon\right)^{1.5-->{2}\right)^\frac{1}{1.5},\\&\quad\approx{a}\times\cos\!\left(\frac{o\!\varepsilon}{2}\right)^2\left(1+\frac{1}{4}\tan\!\left(\frac{o\!\varepsilon}{2}\right)^4\right);\end{align}\,\!
Ramanujan-1914 (#1,#2)
Srinivasa Ramanujan introduced
two different approximations, both from 1914
:\begin{align}1.\;Pr&\approx\pi\Big(3(a+b)-\sqrt{\big(3a+b\big)\big(a+3b\big)}\Big),\\
&\quad=\pi{a}\bigg(6\cos\!\left(\frac{o\!\varepsilon}{2}\right)^2\sqrt{\big(3+\cos\!\left(o\!\varepsilon\right)\big)\big(1+3\cos\!\left(o\!\varepsilon\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\!\left(\frac{o\!\varepsilon}{2}\right)^2\Bigg(1+\frac{3\tan\!\big(\frac{o\!\varepsilon}{2}\big)^4}{10+\sqrt{4-3\tan\!\big(\frac{o\!\varepsilon}{2}\big)^4-->\Bigg);\end{align}\,\!
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{
oε}, results with different ellipticities can be found and compared:
{|class="wikitable"|-! 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|}
External links
- Numericana - Circumference of an ellipse
- Circumference of a circle With interactive applet and animation
Circumference - Wikipedia, the free encyclopedia
The circumference is the distance around a closed curve. Circumference is a kind of perimeter.
Definition: circumference from Online Medical Dictionary
The Online Medical Dictionary is a searchable dictionary of definitions from medicine, science and technology.
Circumference
Circumference of a circle is presented in this interactive lesson from Math Goodies. Learn circumference at your own pace.
Circumference Trusted OnDemand Solutions
Call centre services, including training and management. Company profile, includes full service descriptions.
BBC - Education Scotland - Standard Grade Bitesize Revision - Maths I ...
BBC - Education Scotland - Standard Grade Bitesize Revision - Maths I, Area and Volume, Area and Circumference of a Circle, Introduction. Standard Grade Bitesize is the easy to use ...
BBC - Education Scotland - Standard Grade Bitesize Revision - Maths I ...
BBC - Education Scotland - Standard Grade Bitesize Revision - Maths I, Area and Volume, Area and Circumference of a Circle, The circumference of a circle. Standard Grade Bitesize ...
Circumference | the free film
Circumference is a free film to download and share - sign up here
Circumference -- from Wolfram MathWorld
The perimeter of a circle. For radius r or diameter d=2r, C=2pir=pid, where pi is pi.
circumference - definition of circumference by the Free Online ...
cir·cum·fer·ence (s r-k m f r-ns) n. 1. The boundary line of a circle. 2. a. The boundary line of a figure, area, or object. b. Abbr. c or circ. The length of such a boundary ...
PROJECTOR FILMS: Circumference
Treatment for feature script - The Kiss of Judas ... A modern romance. The world's first feature film that is funded by adverts
