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Circumference is the distance, or perimeter, around a closed curve.

The circumference of a circle has been found to be directly proportional to its diameter, and is represented by the formula:

$c=\pi d$

where the pi symbol π is a dimensionless constant approximately equal to 3.14159.

A common alternate formula is found by substituting the radius, giving:

$c=2\pi r$

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

## 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:

$c=\pi d$

Or, substituting the diameter for the radius:

$c=2\pi r=\pi\cdot{2r}$

where r is the radius and d is the diameter of the circle, and $\pi$ (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...).

If desired, the above circumference formula can be derived without reference to the definition of π by using some integral calculus, as follows:

The upper half of a circle centered at the origin is the graph of the function $f(x)=\sqrt{r^2-x^2}$ , where x runs from $-r$ to $r$ . The circumference (c) of the entire circle can be represented as twice the sum of the lengths of the infinitesimal arcs that make up this half circle. The length of a single infinitesimal part of the arc can be calculated using the Pythagorean formula for the length of the hypotenuse of a right triangle with side lengths $dx$ and $f'(x)dx$ , which gives us $\sqrt{(dx)^2+(f'(x)dx)^2}=\left(\sqrt{1+\left(\frac{df}{dx}\right)^2}\right)dx$ .

Thus the circle circumference can be calculated as follows:

$c=2\int\limits_{-r}^r\sqrt{1+f'(x)^2}dx=2\int\limits_{-r}^r\sqrt{1+\frac{x^2}{r^2-x^2}}dx=2r\int\limits_{-r}^r\frac{dx}{\sqrt{1-x^2}}$

The antiderivative needed to solve this definite integral is the arcsine function:

$c=2r\left[\arcsin\left(\frac{x}{r}\right)\right]_{-r}^{r}=2r\big[\arcsin(1)-\arcsin(-1)\big]=2r\left(\tfrac{\pi}{2}-\big(-\tfrac{\pi}{2}\big)\right)=2\pi r$

Pi (π) is the ratio of the circumference of a circle to its diameter.

## 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 $o\!\varepsilon$ is the ellipse's angular eccentricity,

$o\!\varepsilon=\arccos\left(\frac{b}{a}\right)=2\arctan\left(\sqrt{\frac{a-b}{a+b}}\right)$

\begin{align}\text{E2}[0,90^\circ]&=\text{Integral}'s\text{ divided difference};\\ Pr&=a\times\text{E2}\left[0,90^\circ\right]\quad(\text{perimetric radius});\\ c&=2\pi\times Pr\end{align}

There are many different approximations for the $\mbox{E2}[0,90^\circ]$ divided difference, with varying degrees of sophistication and corresponding accuracy.

In comparing the different approximations, the $\tan^2\left(\frac{o\!\varepsilon}{2}\right)$ based series expansion is used to find the actual value:

\begin{align}\text{E2}[0,90^\circ] &=\cos^2\left(\frac{o\!\varepsilon}{2}\right)\frac{1}{UT}\sum_{TN=1}^{UT=\infty}\binom{0.5}{TN}^2\tan^{4TN}\left(\frac{o\!\varepsilon}{2}\right)\\ &=\cos^2\left(\frac{o\!\varepsilon}{2}\right)\Bigg(1+\frac{\tan^4\left(\frac{o\!\varepsilon}{2}\right)}{4} +\frac{\tan^8\left(\frac{o\!\varepsilon}{2}\right)}{64}\\ &\qquad\qquad\qquad\;\,+\frac{\tan^{12}\left(\frac{o\!\varepsilon}{2}\right)}{256} +\frac{25}{16384}\tan^{16}\left(\frac{o\!\varepsilon}{2}\right) +\cdots\Bigg)\end{align}

### Muir-1883

Probably the most accurate to its given simplicity is Thomas Muir's:
\begin{align}Pr &\approx\left(\frac{a^{1.5}+b^{1.5}}{2}\right)^\frac{1}{1.5}=a\left(\frac{1+\cos^{1.5}\left(o\!\varepsilon\right)}{2}\right)^\frac{1}{1.5}\\ &\quad\approx{a}\times\cos^2\left(\frac{o\!\varepsilon}{2}\right)\left(1+\frac{1}{4}\tan^4\left(\frac{o\!\varepsilon}{2}\right)\right)\end{align}

### Ramanujan-1914 (#1,#2)

Srinivasa Ramanujan introduced two different approximations, both from 1914
\begin{align}1.\;Pr&\approx\frac{3(a+b)-\sqrt{\big(3a+b\big)\big(a+3b\big)}}{2}\\ &\quad=3a\cos^2\left(\frac{o\!\varepsilon}{2}\right)\sqrt{\big(3+\cos(o\!\varepsilon)\big)\big(1+3\cos(o\!\varepsilon)\big)}\end{align}
\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^2\left(\frac{o\!\varepsilon}{2}\right)\Bigg(1+\frac{3\tan^4\big(\frac{o\!\varepsilon}{2}\big)}{10+\sqrt{4-3\tan^4\big(\frac{o\!\varepsilon}{2}\big)}}\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{}, 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

## Circumference of a graph

In graph theory the circumference of a graph refers to the longest cycle contained in that graph.