A central angle is an

angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...

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 Arc length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics and the sciences is a problem in vector calculus and in differential geometry. In the ...

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. It is defined such that one radian is the angle subtended at ...

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

angular distance Angular distance or angular separation is the measure of the angle between the orientation (geometry), orientation of two straight lines, ray (geometry), rays, or vector (geometry), vectors in three-dimensional space, or the central angle subtende ...

. 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 In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions a ...

, then is a straight angle. (In radians, .) Let be the minor arc of the circle between points and , and let be the

radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...

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 Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...

with 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 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 can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...

Formulas

Central angle of a regular polygon

See also

References

External links