Circumference

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:

$c=2\pi r$

Circumference = π × diameter

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{oε}, 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.

External links

Template:Wiktionarypar

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

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