A central angle is an

angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...

whose apex (vertex) is the center O of a circle and whose legs (sides) are

radii In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...

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 ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * ...

is the central angle of a circle of radius one (measured in

radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before tha ...

s). The central angle is also known as the arc's

angular distance Angular distance \theta (also known as angular separation, apparent distance, or apparent separation) is the angle between the two sightlines, or between two point objects as viewed from an observer. Angular distance appears in mathematics (in par ...

. The arc length spanned by a central angle on a sphere is called ''

spherical distance The great-circle distance, orthodromic distance, or spherical distance is the distance along a great circle. It is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a st ...

''. 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 In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid fo ...

, then is a

straight angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...

. (In radians, .) Let be the minor arc of the circle between points and , and let be the

radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...

of the circle. interactive If the central angle is subtended by , then

0^ < \Theta < 180^ \, , \,\, \Theta = \left(  \right) ^=\frac.

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

180^ < \Theta < 360^ \, , \,\, \Theta = \left( 360 - \frac \right) ^=2\pi-\frac.

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

regular polygon In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...

with sides has a

circumscribed circle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

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 radii to two adjacent vertices. The measure of this angle is

2\pi/n.

References

External links

* interactive *{{cite web, url=http://www.mathopenref.com/arccentralangletheorem.html, title=Central Angle Theorem, publisher =Math Open Reference, year=2009, accessdate=December 30, 2013 interactive
Inscribed and Central Angles in a Circle
Angle Circles Elementary geometry

Angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...

Formulas

Central angle of a regular polygon

See also

References

External links