Library/Geometry/Plane geometry/Circles/Central angle. Length of arc and circumference

Central angle. Length of arc and circumference

Overview
Important

A central angle is an angle with its vertex at the center of a circle and its sides (rays) intersecting the circle at two points, creating an arc. The length of this arc is proportional to the size of the angle. The circumference is the total length around the circle, given by C=2πrC = 2\pi r where rr is the radius.

Important properties

  • The measure of a central angle (in degrees) divided by 360 equals the fraction of the circle's circumference that the arc covers.

  • Arc length formula: If the central angle is θ\theta degrees and the radius is rr, then arc length =θ360×2πr= \frac{\theta}{360} \times 2\pi r.

  • If the angle is in radians, arc length =rθ= r\theta.

  • The circumference is the arc length when the central angle is 360∘360^\circ (a full circle).