Circumference

The circumference (from Latin circumferentia, meaning "carrying around") of a closed curve or circular object is the linear distance around its edge.^[1] The circumference of a circle is of special importance in geometry and trigonometry. Informally "circumference" may also refer to the edge itself rather than to the length of the edge. Circumference is a special case of perimeter: the perimeter is the length around any closed figure, but conventionally "perimeter" is typically used in reference to a polygon while "circumference" typically refers to a continuously differentiable curve.

Circumference of a circle

Circle illustration with circumference (C) in black, diameter (D) in cyan, radius (R) in red, and centre or origin (O) in magenta. Circumference = π × diameter = 2 × π × radius.

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

When a circle's diameter is 1, its circumference is π.

When a circle's radius is 1—called a unit circle—its circumference is 2π.

Relationship with Pi

The circumference of a circle relates to one of the most 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  A000796). Pi is defined as the ratio of a circle's circumference C to its diameter d:

$\pi = \frac{C}{d}$

Or, equivalently, as the ratio of the circumference to twice the radius. The above formula can be rearranged to solve for the circumference:

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

The use of the mathematical constant π is ubiquitous in mathematics, engineering, and science. The constant ratio of circumference to radius ${C}/{r} = 2\pi$ also has many uses in mathematics, engineering, and science. These uses include but are not limited to radians, computer programming, and physical constants. The Greek letter τ (tau) is sometimes used to represent this constant, but is not generally accepted as proper notation.

Circumference of an ellipse

The circumference of an ellipse can be expressed in terms of the complete elliptic integral of the second kind.

Circumference of a graph

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

See also

• Arclength
• Area
• Caccioppoli set
• Isoperimetric inequality
• Pythagorean Theorem
• Volume

References

1. San Diego State University (2004). "Perimeter, Area and Circumference" (PDF). Addison-Wesley.

External links

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