Editing Central angle

{{Short description|Measure of two radii meeting}}
[[Image:Sector central angle arc.svg|thumb|right|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 [[radius|radii]] intersecting the circle in two distinct points A and B. Central angles are [[subtend]]ed by an [[Arc (geometry)|arc]] between those two points, and the [[arc length]] is the central angle of a circle of radius one (measured in [[radian]]s).<ref name=Oxford>{{cite web | url=http://web.cortland.edu/matresearch/OxfordDictionaryMathematics.pdf |title=Oxford Concise Dictionary of Mathematics, Central Angle| first1=C.|last1=Clapham|first2=J.|last2=Nicholson | publisher =Addison-Wesley | year =2009|page=122|accessdate=December 30, 2013}}</ref> 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 {{math|Θ}} is  {{math|0° &lt; Θ &lt; 360°}}  or   {{math|0 &lt; Θ &lt; 2π}} (radians). When defining or drawing a central angle, in addition to specifying the points {{mvar|A}} and {{mvar|B}}, one must specify whether the angle being defined is the convex angle (&lt;180°) or the reflex angle (&gt;180°). Equivalently, one must specify whether the movement from point {{mvar|A}} to point {{mvar|B}} is clockwise or counterclockwise.

==Formulas==
If the intersection points {{mvar|A}} and {{mvar|B}} of the legs of the angle with the circle form a [[diameter]], then {{math|1=Θ = 180°}} is a [[straight angle]]. (In radians, {{math|1=Θ = π}}.)

Let {{math|''L''}} be the '''minor arc''' of the circle between points {{mvar|A}} and {{mvar|B}}, and let {{mvar|R}} be the [[radius]] of the circle.<ref>{{cite web|url=http://www.mathopenref.com/circlecentral.html|title=Central angle (of a circle)| publisher =Math Open Reference|year=2009|accessdate=December 30, 2013}} interactive</ref>
[[Image:Angle central convex.svg|frame|right|Central angle. Convex. Is subtended by minor arc {{math|''L''}}]]
If the central angle {{math|Θ}} is subtended by {{math|''L''}}, then 
<math display="block"> 0^{\circ} < \Theta < 180^{\circ} \, , \,\, \Theta = \left( {\frac{180L}{\pi R}} \right) ^{\circ}=\frac{L}{R}.</math>

{{math proof|title=Proof (for degrees)|proof= The [[circumference]] of a circle with radius {{mvar|R}} is {{math|2π''R''}}, and the minor arc {{math|''L''}} is the ({{sfrac|Θ|360°}}) proportional part of the whole circumference (see [[Arc (geometry)|arc]]). So: 
<math display="block">L=\frac{\Theta}{360^{\circ}} \cdot 2 \pi R \, \Rightarrow  \, \Theta = \left( {\frac{180L}{\pi R}} \right) ^{\circ}.</math>}}

[[Image:Angle central reflex.svg|frame|right|Central angle. Reflex. Is ''not'' subtended by {{math|''L''}}]]
{{math proof|title=Proof (for radians)|proof= The [[circumference]] of a circle with radius {{mvar|R}} is {{math|2π''R''}}, and the minor arc {{math|''L''}} is the ({{sfrac|Θ|2π}}) proportional part of the whole circumference (see [[Arc (geometry)|arc]]). So
<math display="block">L=\frac{\Theta}{2 \pi} \cdot 2 \pi R \, \Rightarrow  \, \Theta = \frac{L}{R}.</math>}}

If the central angle {{math|Θ}} is '''not''' subtended by the minor arc {{math|''L''}}, then {{math|Θ}} is a reflex angle and
<math display="block"> 180^{\circ} < \Theta < 360^{\circ} \, , \,\, \Theta = \left( 360 - \frac{180L}{\pi R} \right) ^{\circ}=2\pi-\frac{L}{R}.</math>

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

==Central angle of a regular polygon==

A [[regular polygon]] with {{math|''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 radii to two adjacent vertices. The measure of this angle is <math>2\pi/n.</math>

== See also ==
*[[Chord (geometry)]]
*[[Inscribed angle]]
*[[Great-circle navigation]]

==References==
{{reflist}}

==External links==
*{{cite web|url=http://www.mathopenref.com/circlecentral.html|title=Central angle (of a circle)| publisher =Math Open Reference|year=2009|accessdate=December 30, 2013}} 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
*[http://www.cut-the-knot.org/Curriculum/Geometry/InscribedAngle.shtml Inscribed and Central Angles in a Circle]

[[Category:Angle]]
[[Category:Circles]]
[[Category:Elementary geometry]]
[[Category:Geometric centers|Angle]]