Overview

An arc is a continuous curved segment that lies on the circumference of a circle. In elementary geometry an arc is formed by two points on a circle and the portion of the circle between them. Because it follows a circle, an arc has constant curvature and is shorter than the corresponding major arc that goes the long way around the same two endpoints.

Definition and notation

The arc determined by two points A and B on a circle is commonly called "arc AB." In handwritten and printed mathematics this is often indicated by placing a small curved mark over the two letters (read as "arc AB"). The two points A and B also define the chord AB (the straight segment joining them), and the central angle subtending the arc: the angle at the circle's center between the radii to A and B.

Properties and formulas

The length of an arc depends on the circle's radius r and the measure of the central angle θ that subtends it. When θ is measured in radians the arc length s is given by the simple relation s = r·θ. If θ is given in degrees, one may convert by θ (radians) = π·θ°/180, so the length becomes s = π·r·θ°/180. Arc length is proportional to the central angle and to the radius.

Arcs are classified by their size relative to the circle:

  • Minor arc: the shorter arc between two points (central angle less than 180°).
  • Major arc: the longer arc connecting the same points (central angle greater than 180°).
  • Semicircle: an arc determined by endpoints that are diametrically opposite (central angle 180°).

Other related regions include a sector (the area bounded by two radii and the arc) and a segment (the region between an arc and its chord).

History and applications

Arcs are fundamental in classical geometry and trigonometry, studied since ancient Greek mathematics as part of circle theory. Today they appear across engineering, architecture, computer graphics, and navigation: arcs model curved beams and arches, define paths for gears and wheels, and are used in drawing and CAD tools. Practical computations typically use the s = r·θ relation or numerical approximation when shapes are not perfect circles.

Notable facts and distinctions

When discussing arc length it is important to specify the unit and the angle measure. The same pair of endpoints determines two distinct arcs (minor and major) unless they are endpoints of a diameter. For formal study of curves beyond circles, the term "arc" can refer to any continuous curve segment, but in standard Euclidean geometry it most often means a circular arc. For more on the circle and related constructions, see circle.