circumference

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

Circumference = π × diameter

Circle

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

Or, substituting the radius for the diameter:

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 is the ellipse's angular eccentricity,

Failed to parse (unknown function\begin): \begin{align}\mbox{E2}\left[0,90^\circ\right]&= \mbox{Integral}'s\mbox{ divided difference};\\ Pr&=a\times\mbox{E2}\left[0,90^\circ\right] \quad(\mbox{perimetric radius});\\ c&=2\pi\times Pr.\end{align}\,\!

There are many different approximations for the divided difference, with varying degrees of sophistication and corresponding accuracy.

In comparing the different approximations, the based series expansion is used to find the actual value:

Failed to parse (unknown function\begin): \begin{align}\mbox{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},\\ &=\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's:

Failed to parse (unknown function\begin): \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

Failed to parse (unknown function\begin): \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}\,\!

Failed to parse (unknown function\begin): \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:

External links

• Numericana - Circumference of an ellipse
• Circumference of a circle With interactive applet and animation

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