Circumference

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

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The circumference of a circle can be calculated from its diameter using the formula:

c = \pi \cdot d

Or, substituting the 2 radius for the diameter:

c = 2\pi \cdot r

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...

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\!\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);\,\!

\operatorname{E2}\left[0,90^\circ\right]= \mbox{Integral}'s\mbox{ divided difference};

  Pr=a\times\operatorname{E2}\left[0,90^\circ\right] \quad(\mbox{perimetric radius});\,\!
c=2\pi\times Pr.\,\!

There are many different approximations for \operatorname{E2}\left[0,90^\circ\right], 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:

\operatorname{E2}\left[0,90^\circ\right] =\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},\,\!
\quad = \cos\!\left(\frac{o\!\varepsilon}{2}\right)^2\left[ 1+\frac{1}{4}\tan\!\left(\frac{o\!\varepsilon}{2}\right)^4 +\frac{1}{64}\tan\!\left(\frac{o\!\varepsilon}{2}\right)^8 +\frac{1}{256}\tan\!\left(\frac{o\!\varepsilon}{2}\right)^{12}+... \right];\,\!

Probably the most accurate to its given simplicity is Thomas Muir's:
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},\,\!
\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];\,\!

Srinivasa Ramanujan introduced two different approximations, both from 2007
  1.\ Pr \approx \pi \left[3(a+b) - \sqrt{(3a+b)(a+3b)}\right];\,\!
=\pi a\left[6\cos\!\left(\frac{o\!\varepsilon}{2}\right)^2 \sqrt{(3+\cos\!\left(o\!\varepsilon\right))(1+3\cos\!\left(o\!\varepsilon\right))}\right];\,\!
  2.\ Pr \approx\frac{1}{2}\left[a+b\right]\left[1+\frac{3\left[\frac{a-b}{a+b}\right]^2}{10+\sqrt{4-3\left[\frac{a-b}{a+b}\right]^2}}\right];\,\!
    =a\times \cos\!\left(\frac{o\!\varepsilon}{2}\right)^2\left[1+\frac{3\tan\!\left(\frac{o\!\varepsilon}{2}\right)^4} {10+\sqrt{4-3\tan\!\left(\frac{o\!\varepsilon}{2}\right)^4}}\right];\,\!
The second equation is 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 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

Look up circumference in Wiktionary, the free dictionary.
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