# Circumference

Circumference is the linear distance around the outside of a closed curve or circular object.[1] The circumference of a circle is of special importance to geometric and trigonometric concepts. However circumference may also describe the outside of elliptical closed curves. Circumference is a special example of perimeter.[2]

## Circumference of a circle

Circle illustration Circumference = π × diameter = π × 2 × radius

The circumference of a circle is the length around it. The term is used when measuring physical objects, as well as when considering abstract geometric forms.

### Relationship with Pi

When a circle's diameter is 1 unit, its circumference is π units.
When a circle's radius is 1 unit, its circumference is 2π.

The circumference of a circle relates to one of the most fundamental and important mathematical constants in all of mathematics. This constant pi, is represented by the Greek letter π. The numerical value of π is 3.14159 26535 89793 ... (see ), and is defined by two proportionality constants.[3] The first constant is the ratio of a circle's circumference to its diameter and equals π. While the second constant is the ratio of the diameter and two times the radius and is used as to convert the diameter to radius in the same ratio as the first, π.[4] Both proportionality constants combine in respect with circumference c, diameter d, and radius r to become:

${C}=\pi\cdot{d}={2}\cdot\pi\cdot{r}.\!$

The use of the mathematical constant π is ubiquitous in mathematics, engineering, and science. While the constant ratio of circumference to radius ${C}/{r} = 2\cdot\pi$ also has many uses in mathematics, engineering, and science, it is not formally named. These uses include but are not limited to radians, computer programming, and physical constants.

## Circumference of an ellipse

The circumference of an ellipse is more difficult to calculate than that of a circle.[5] It involves higher order mathematics. The circumference and diameters of an ellipse are not related in the simple, linear manner of a circle. The exact measurement requires using advanced calculus to find the complete elliptic integral of the second kind.[6] This can be achieved either via numerical integration or binomial series expansions.

## Circumference of a graph

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

1. ^ Addison-Wesley (2004). "Perimeter, Area and Circumference". sdsu.edu. Unknown parameter |name= ignored (|author= suggested) (help)
5. ^ University of Denver; J. B. Calvert (May 6, 2002). "Definition of the Ellipse". du.edu. Unknown parameter |name= ignored (|author= suggested) (help)