CIRCUMFERENCE

The 'circumference' is the distance around a closed curve. Circumference is a kind of perimeter.
Circumference = π × diameter


Contents
Circle
Ellipse
Muir-1883
Ramanujan-1914 (#1,#2)
External links

Circle


The circumference of a circle can be calculated from its diameter using the formula:
:c=picdot{d}.,!
Or, substituting the radius for the diameter:
:c=2picdot{r}=picdot{2r},,!
where ''r'' is the radius and ''d'' is the diameter of the circle, and π (the Greek letter pi) is the 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 and semi-minor axes, respectively, and o! arepsilon,! is the ellipse's angular eccentricity,
o! arepsilon=rccos!left( rac{b}{a}
ight)=2rctan!left(!sqrt{ rac{a-b}{a+b}},
ight);,!
egin{align}mbox{E2}left[0,90^circ
ight]&= mbox{Integral}'smbox{ divided difference};\
Pr&=a imesmbox{E2}left[0,90^circ
ight] quad(mbox{perimetric radius});\
c&=2pi imes Pr.end{align},!
There are many different approximations for the mbox{E2}left[0,90^circ
ight] divided difference, with varying degrees of sophistication and corresponding accuracy.
In comparing the different approximations, the an!left( rac{o! arepsilon}{2}
ight)^2,! based series expansion is used to find the actual value:
egin{align}mbox{E2}left[0,90^circ
ight]
&=cos!left( rac{o! arepsilon}{2}
ight)^2 rac{1}{UT}sum_{TN=1}^{UT=infty}{.5choose{}TN}^2 an!left( rac{o! arepsilon}{2}
ight)^{4TN},\
&=cos!left( rac{o! arepsilon}{2}
ight)^2Bigg(1+ rac{1}{4} an!left( rac{o! arepsilon}{2}
ight)^4
+ rac{1}{64} an!left( rac{o! arepsilon}{2}
ight)^8\ &qquadqquadqquad;,+ rac{1}{256} an!left( rac{o! arepsilon}{2}
ight)^{12}
+ rac{25}{16384} an!left( rac{o! arepsilon}{2}
ight)^{16}
+...Bigg);end{align},!
Muir-1883

:Probably the most accurate to its given simplicity is Thomas Muir's:
::egin{align}Pr
&pproxleft( rac{a^{1.5}+b^{1.5}}{2}
ight)^ rac{1}{1.5}=aleft( rac{1+cos!left(o! arepsilon
ight)^{1.5}}{2}
ight)^ rac{1}{1.5},\
&quadpprox{a} imescos!left( rac{o! arepsilon}{2}
ight)^2left(1+ rac{1}{4} an!left( rac{o! arepsilon}{2}
ight)^4
ight);end{align},!
Ramanujan-1914 (#1,#2)

:Srinivasa Ramanujan introduced ''two'' different approximations, both from 1914
::egin{align}1.;Pr&pproxpiBig(3(a+b)-sqrt{ig(3a+big)ig(a+3big)}Big),\
&quad=pi{a}igg(6cos!left( rac{o! arepsilon}{2}
ight)^2sqrt{ig(3+cos!left(o! arepsilon
ight)ig)ig(1+3cos!left(o! arepsilon
ight)ig)}igg);end{align},!
::egin{align}2.;Pr&pprox rac{1}{2}Big(a+bBig)Bigg(1+ rac{3ig( rac{a-b}{a+b}ig)^2}{10+sqrt{4-3ig( rac{a-b}{a+b}ig)^2}}Bigg);\
&quad=a imescos!left( rac{o! arepsilon}{2}
ight)^2Bigg(1+ rac{3 an!ig( rac{o! arepsilon}{2}ig)^4}{10+sqrt{4-3 an!ig( rac{o! arepsilon}{2}ig)^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:
b Pr Ramanujan-#2 Ramanujan-#1 Muir
9975 '9987.50391 11393'  '9987.50391 11393'  '9987.50391 11393'  '9987.50391 113'89
9966 '9983.00723 73047' '9983.00723 73047' '9983.00723 73047' '9983.00723 730'34
9950 '9975.01566 41666' '9975.01566 41666' '9975.01566 41666' '9975.01566 416'04
9900 '9950.06281 41695' '9950.06281 41695' '9950.06281 41695' '9950.06281 4'0704
9000 '9506.58008 71725' '9506.58008 71725' '9506.58008' 67774 '9506.5'7894 84209
8000 '9027.79927 77219' '9027.79927 77219' '9027.7992'4 43886 '9027.7'7786 62561
7500 '8794.70009 24247' '8794.70009 2424'0 '8794'.69994 52888 '8794'.64324 65132
6667 '8417.02535 37669' '8417.02535 37'460 '8417.02'428 62059 '841'6.81780 56370
5000 '7709.82212 59502' '7709.82212' 24348 '7709.8'0054 22510 '770'8.38853 77837
3333 '7090.18347 61693' '7090.183'24 21686 '70'89.94281 35586 '70'83.80287 96714
2500 '6826.49114 72168' '6826.4'8944 11189 '682'5.75998 22882 '68'14.20222 31205
1000 '6468.01579 36089' '646'7.94103 84016 '646'2.57005 00576 '64'31.72229 28418
 100 '6367.94576 97209' '636'6.42397 74408 '63'46.16560 81001 '63'03.80428 66621
  10 '6366.22253 29150' '636'3.81341 42880 '63'40.31989 06242 '6'299.73805 61141
   1 '6366.19804 50617' '636'3.65301 06191 '63'39.80266 34498 '6'299.60944 92105
iota '6366.19772 36758' '636'3.63636 36364 '63'39.74596 21556 '6'299.60524 94744

External links



★ Numericana - Circumference of an ellipse

★ Circumference of a circle With interactive applet and animation

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