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

Circumference of a circle

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

$c=\backslash pi\backslash cdot\{d\}.\backslash ,\backslash !$

Or, substituting the diameter for the radius:

$c=2\backslash pi\backslash cdot\{r\}=\backslash pi\backslash cdot\{2r\},\backslash ,\backslash !$

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

Thus the circle circumference can be calculated as

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

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

$c\; =\; 2r\; \backslash bigarcsin(\backslash frac\{x\}\{r\})\; \backslash big\_\{-r\}^\{r\}\; =\; 2r\; \backslash bigarcsin(1)-arcsin(-1)\; \backslash big=\; 2r(\backslash tfrac\{\backslash pi\}\{2\}-(-\backslash tfrac\{\backslash pi\}\{2\}))\; =\; 2\backslash pi\; r.$

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

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

There are many different approximations for the $\backslash mbox\{E2\}\backslash left3$ divided difference, with varying degrees of sophistication and corresponding accuracy.