Definitions

# Circumference

[ser-kuhm-fer-uhns]
The circumference is the distance around a closed curve. Circumference is a kind of perimeter.

## Circumference of a circle

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

$c=picdot\left\{d\right\}.,!$

Or, substituting the diameter for the radius:

$c=2picdot\left\{r\right\}=picdot\left\{2r\right\},,!$

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...).

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 $f\left(x\right) = sqrt\left\{r^2-x^2\right\},$ 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 rectangular triangle with side lengths dx and f'(x)dx, which gives us $sqrt\left\{\left(dx\right)^2+\left(f\text{'}\left(x\right)dx\right)^2\right\} = left\left(sqrt\left\{1+f\text{'}\left(x\right)^2\right\} right\right) dx.$

Thus the circle circumference can be calculated as

$c = 2 int_\left\{-r\right\}^r sqrt\left\{1+f\text{'}\left(x\right)^2\right\}dx$ = $2 int_\left\{-r\right\}^r sqrt\left\{1+frac\left\{x^2\right\}\left\{r^2-x^2\right\}\right\}dx$ = $2 int_\left\{-r\right\}^r sqrt\left\{frac\left\{1\right\}\left\{1-frac\left\{\left\{x\right\}^2\right\}\left\{\left\{r\right\}^2\right\}\right\}\right\}dx$

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

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

## 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,

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

begin\left\{align\right\}mbox\left\{E2\right\}left\left[0,90^circright\right]&= mbox\left\{Integral\right\}\text{'}smbox\left\{ divided difference\right\}; Pr&=atimesmbox\left\{E2\right\}left\left[0,90^circright\right] quad\left(mbox\left\{perimetric radius\right\}\right); c&=2pitimes Pr.end\left\{align\right\},!

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

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

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

### Muir-1883

Probably the most accurate to its given simplicity is Thomas Muir's:
begin\left\{align\right\}Pr

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

Srinivasa Ramanujan introduced two different approximations, both from 1914
begin\left\{align\right\}1.;Pr&approxpiBig\left(3\left(a+b\right)-sqrt\left\{big\left(3a+bbig\right)big\left(a+3bbig\right)\right\}Big\right),

begin\left\{align\right\}2.;Pr&approxfrac\left\{1\right\}\left\{2\right\}Big\left(a+bBig\right)Bigg\left(1+frac\left\{3big\left(frac\left\{a-b\right\}\left\{a+b\right\}big\right)^2\right\}\left\{10+sqrt\left\{4-3big\left(frac\left\{a-b\right\}\left\{a+b\right\}big\right)^2\right\}\right\}Bigg\right);

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