In geometry, a central angle is an angle whose apex (vertex) is the center of a circle and whose legs (sides) are radii intersecting the circle in two distinct points and . 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 (measured in radians). The central angle is also known as the arc's angular distance. The arc length spanned by a central angle on a sphere is called spherical distance.

The size of a central angle is or (radians). 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 (180°) or the reflex angle (180°). 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. (In radians, .)

Let be the minor arc of the circle between points and , and let be the radius of the circle. If the central angle is subtended by

If the central angle is not subtended by the minor arc , then

If a tangent at and a tangent at intersect at the exterior point , then denoting the center as , the angles (convex) and are supplementary (sum to 180°).

Central angle of a regular polygon

A regular polygon with sides has a circumscribed circle

See also

• Chord (geometry)

• Inscribed angle

• Great-circle navigation

References

External links

• interactive

• interactive

• Inscribed and Central Angles in a Circle