# Circumference

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

## Contents

### Circumference of a circle

The circumference of a circle is the length around it. 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 defined as the ratio of the circumference of the circle to its diameter (the numerical value of pi is 3.141 592 653 589 793...).

### Circumference of an 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 $\alpha\,\!$ is the ellipse's angular eccentricity,

$\alpha=\arccos\!\left(\frac{b}{a}\right)=2\arctan\!\left(\!\sqrt{\frac{a-b}{a+b}}\,\right);\,\!$

\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 $\mbox{E2}\left[0,90^\circ\right]$ divided difference, with varying degrees of sophistication and corresponding accuracy.

In comparing the different approximations, the $\tan^2\!\left(\frac{\alpha}{2}\right)\,\!$ (also known as "n", the third flattening of the ellipse) based series expansion is used to find the actual value:

\begin{align}\mbox{E2}\left[0,90^\circ\right] &=\cos^2\!\left(\frac{\alpha}{2}\right)\frac{1}{UT}\sum_{TN=1}^{UT=\infty}{.5\choose{}TN}^2\tan^{4TN}\!\left(\frac{\alpha}{2}\right),\\ &=\cos^2\!\left(\frac{\alpha}{2}\right)\Bigg(1+\frac{1}{4}\tan^4\!\left(\frac{\alpha}{2}\right) +\frac{1}{64}\tan^8\!\left(\frac{\alpha}{2}\right)\\ &\qquad\qquad\qquad\;\,+\frac{1}{256}\tan^{12}\!\left(\frac{\alpha}{2}\right) +\frac{25}{16384}\tan^{16}\!\left(\frac{\alpha}{2}\right) +...\Bigg);\end{align}\,\!