Central angle
A central angle is an angle whose apex is the center O of a circle and whose legs are radii intersecting the circle in two distinct points A and B. Central angles are subtended by an arc between those two points, and the arc length is the central angle of a circle of radius one. The central angle is also known as the arc's angular distance.
The size of a central angle is or . When defining or drawing a central angle, in addition to specifying the points and, one must specify whether the angle being defined is the convex angle or the reflex angle. Equivalently, one must specify whether the movement from point to point is clockwise or counterclockwise.
Formulas
If the intersection points and of the legs of the angle with the circle form a diameter, then is a straight angle.Let be the minor arc of the circle between points and, and let be the radius of the circle.
| Central angle. Convex. Is subtended by minor arc |
If the central angle is subtended by, then
| Central angle. Reflex. Is not subtended by |
If the central angle is not subtended by the minor arc, then is a reflex angle and
If a tangent at and a tangent at intersect at the exterior point, then denoting the center as, the angles and are supplementary.