Circumference
Encyclopedia : C : CI : CIR : Circumference
The circumference is the distance around a closed curve. Circumference is a kind of perimeter.
Circle
The circumference of a circle can be calculated from its diameter using the formula:[c = \pi d]
Or, substituting the radius for the diameter:
[c = 2\pi 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...
Ellipse
The circumference of an ellipse is more problematical, 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 [e\,\!] is the ellipse's eccentricity,
[O\!\!E = \arcsin\!\left\=\arccos\!\left\\right\}\quad (\mbox\ modular\ angle\mbox\ angular\ eccentricity\ );\,\!]
[\operatorname\left[0,90^circright]= \mbox's\mbox;]
- [Pr=a\times\operatorname\left[0,90^circright] \quad(\mbox);\,\!]
- :[c=2\pi\times Pr.\,\!]
In comparing the different approximations, the [tan\!\left\\right\}^2\,\!] based series expansion is used to find the actual value:
| [\operatorname\left[0,90^circright]] | [=cos\!\left\\right\}^2 \frac\sum_^^2tan\!\left\\right\}^,\,\!] |
| [\quad = cos\!\left\\right\}^2\left[1+fractan!leftright}^4+fractan!leftright}^8+fractan!leftright}^+...right]\,\!] |
Muir-1883
- Probably the most accurate to its given simplicity is Thomas Muir's:
- :[Pr \approx \left[frac+b^}right]^\frac =a\left[frac^}right]^\frac,\,\!]
- :::[\approx a\times cos\!\left\\right\}^2\left[1+fractan!leftright}^4right]\,\!]
Ramanujan-1914 (#1,#2)
- Srinivasa Ramanujan introduced two different approximations, both from 1914:
- : [1.\ Pr \approx \frac\left[3(a+b) - sqrtright]\,\!]
- :::[=\fraca\left[6cos!leftright}^2 sqrt)(1+3cos!left)}right]\,\!]
- : [2.\ Pr \approx\frac\left[a+bright]\left[1+fracright]^2}right]^2}}right]\,\!]
- :: [=a\timescos\!\left\\right\}^2\left[1+fracright}^4}right}^4}}right]\,\!]
- The second equation is by far the better of the two, and may be the most accurate approximation known.
| 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 |
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