Central angle

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title: "Central angle" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["angle", "circles", "elementary-geometry", "geometric-centers"] description: "Measure of two radii meeting" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Central_angle" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Measure of two radii meeting ::

::figure[src="https://upload.wikimedia.org/wikipedia/commons/0/08/Sector_central_angle_arc.svg" caption="Angle AOB is a central angle"] ::

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. 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 Θ is {{math|0° 180°). Equivalently, one must specify whether the movement from point A to point B is clockwise or counterclockwise.

Formulas

If the intersection points A and B of the legs of the angle with the circle form a diameter, then is a straight angle. (In radians, .)

Let L be the minor arc of the circle between points A and B, and let R be the radius of the circle. ::figure[src="https://upload.wikimedia.org/wikipedia/commons/3/37/Angle_central_convex.svg" caption="''L''}}"] ::

If the central angle Θ is subtended by L, then 0^{\circ}

L=\frac{\Theta}{360^{\circ}} \cdot 2 \pi R , \Rightarrow , \Theta = \left( {\frac{180L}{\pi R}} \right) ^{\circ}.}}

::figure[src="https://upload.wikimedia.org/wikipedia/commons/0/08/Angle_central_reflex.svg" caption="''L''}}"] ::

L=\frac{\Theta}{2 \pi} \cdot 2 \pi R , \Rightarrow , \Theta = \frac{L}{R}.}}

If the central angle Θ is not subtended by the minor arc L, then Θ is a reflex angle and 180^{\circ}

If a tangent at A and a tangent at B intersect at the exterior point P, then denoting the center as O, the angles ∠BOA (convex) and ∠BPA are supplementary (sum to 180°).

Central angle of a regular polygon

A regular polygon with n sides has a circumscribed circle upon which all its vertices lie, and the center of the circle is also the center of the polygon. The central angle of the regular polygon is formed at the center by the circumradii to two adjacent vertices. The measure of this angle is 2\pi/n.

References

References

1. (2009). "Oxford Concise Dictionary of Mathematics, Central Angle". Addison-Wesley.
2. (2009). "Central angle (of a circle)". Math Open Reference.

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::