Circumference: length around a closed curve, especially a circle

Formulas and related quantities

For a circle, the circumference can be expressed in terms of the diameter or the radius. The standard formulas are:

• C = π d, where d is the diameter.
• C = 2 π r, where r is the radius.
• For an arc of a circle with central angle θ measured in radians: arc length s = r θ.

Characteristics and calculation

Circumference depends only on linear measures of the shape (diameter or radius for a circle) and the mathematical constant π, the ratio of circumference to diameter for any circle. For non-circular closed curves the circumference is computed as an arc-length integral; practical measurement may use approximation, segmentation, or numerical integration methods.

History and noteworthy facts

Understanding of circumference and the constant π dates back to ancient civilizations that approximated the ratio between a circle's circumference and its diameter. Over time techniques improved from geometric constructions to analytic and numerical methods. A notable mathematical fact is that π is an irrational number, so the exact circumference cannot be expressed as a ratio of integers.

Uses, examples and distinctions

Circumference appears in engineering (wheel and gear design), navigation, construction, and any situation requiring a length around a curve. It is distinct from area (which measures enclosed region) and from the generic term perimeter (which applies to polygons and other shapes). Simple examples: a wheel with radius 0.5 m has circumference C = 2π(0.5) = π m; a circle with diameter 10 cm has C = π·10 = 10π cm.

Practical notes

When computing arc lengths or manufacturing curved parts, convert angles to radians for use with s = rθ, and use sufficient numerical precision for π when high accuracy is required. For more on related concepts, see pages about closed curves, the circle, perimeter (perimeter) and diameter.