A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B thereby subtending an arc between those two points whose angle is (by definition) equal to that of the central angle itself. It is also known as the arc segment's angular distance.
When defining or drawing a central angle, in addition to specifying the points A and B, one must specify and/or denote whether the angle being defined is the convex angle (<180°) or the reflex angle (>180°).
- The size of a central angle Θ is: 0°<Θ<360° оr 0<Θ<2π (radians)
- If the intersection points A and B of the legs of the angle with the circle form a diameter, then Θ=180° is a straight angle. (In radians, Θ=π.)
|Central angle. Convex. Includes minor arc L|
- If the central angle Θ includes L, then
|Central angle. Reflex. Does not include L|
- If the central angle Θ does not include the minor arc L, then the Θ is a reflex angle and:
- If a tangent at A and a tangent at B intersect at the exterior point P, then denoting the centre as O, the angles ∠BOA (convex) and ∠BPA are supplementary (sum to 180 °).
- Clapham, C.; Nicholson, J. (2009). "Oxford Concise Dictionary of Mathematics, Central Angle" (PDF). Addison-Wesley. p. 122. Retrieved December 2013.
- "Central angle (of a circle)". Math Open Reference. 2009. Retrieved December 2013. interactive