Circumference is the linear distance around the edge of a closed curve or circular object.^{[1]} The circumference of a circle is of special importance in geometry and trigonometry. However "circumference" may also refer to the edge of elliptical closed curve. Circumference is a special case of perimeter in that the perimeter is typically around a polygon while circumference is around a closed curve.

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 radius is 1, its circumference is 2π.

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

Relationship with Pi[edit]

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 letterπ. The numerical value of π is 3.14159 26535 89793 ... (see A000796), and is defined by two proportionality constants. 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. Both proportionality constants combine in respect with circumference c, diameter d, and radius r to become:

The use of the mathematical constant π is ubiquitous in mathematics, engineering, and science. While the constant ratio of circumference to radius 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.