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Circumference
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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
  • Area--- = \pi r^2-->


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\{e ight\}=\arccos\!\left\{ rac{b}{a} ight\}\quad (\mbox{the }\ modular\ angle\mbox{ or }\ angular\ eccentricity\ );\,\!

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

:  Pr=a imes\operatorname{E2}\left[0,90^\circ ight] \quad(\mbox{perimetric radius});\,\!

::c=2\pi imes Pr.\,\!

There are many different Approximation s for \operatorname{E2}\left[0,90^\circ ight], with varying degrees of sophistication and corresponding accuracy.

In comparing the different approximations, the tan\!\left\{ rac{O\!\!E}{2} ight\}^2\,\! based series expansion is used to find the actual value:


Muir-1883

:Probably the most accurate to its given simplicity is Thomas Muir's :
::Pr \approx \left[ rac{a^{1.5}+b^{1.5}}{2} ight]^ rac{1}{1.5} =a\left[ rac{1+\cos\!\left\{O\!\!E ight\}^{1.5}}{2} ight]^ rac{1}{1.5},\,\!
::::\approx a imes cos\!\left\{ rac{O\!\!E}{2} ight\}^2\left[1+ rac{1}{4}tan\!\left\{ rac{O\!\!E}{2} ight\}^4 ight];\,\!


Ramanujan-1914 (#1,#2)

: Srinivasa Ramanujan introduced ''two'' different approximations, both from 1914:
::  1.\ Pr \approx rac{1}{2}\left - \sqrt{(3a+b)(a+3b)} ight ;\,\!
::::= rac{1}{2}a\left[6\cos\!\left\{ rac{O\!\!E}{2} ight\}^2 sqrt{(3+\cos\!\left\{O\!\!E ight\})(1+3\cos\!\left\{O\!\!E ight\})} ight];\,\!

::  2.\ Pr \approx rac{1}{2}\left[a+b ight]\left[1+ rac{3\left[ rac{a-b}{a+b} ight]^2}{10+\sqrt{4-3\left[ rac{a-b}{a+b} ight]^2}} ight];\,\!
:::    =a imes
cos\!\left\{ rac{O\!\!E}{2} ight\}^2\left[1+ rac{3 an\!\left\{ rac{O\!\!E}{2} ight\}^4}
{10+\sqrt{4-3 an\!\left\{ rac{O\!\!E}{2} ight\}^4}} ight];\,\!

: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{''Œ''}, results with different ellipticities can be found and compared:


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