Login
Login

central angle

{{Short description|Measure of two radii meeting}}

Image:Sector central angle arc.svg

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).{{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}} 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° < Θ < 360°}} or {{math|0 < Θ < 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 (<180°) or the reflex angle (>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.{{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

Image:Angle central convex.svg

If the central angle {{math|Θ}} is subtended by {{math|L}}, then

0^{\circ} < \Theta < 180^{\circ} \, , \,\, \Theta = \left( {\frac{180L}{\pi R}} \right) ^{\circ}=\frac{L}{R}.

{{math proof|title=Proof (for degrees)|proof= The circumference of a circle with radius {{mvar|R}} is {{math|2πR}}, and the minor arc {{math|L}} is the ({{sfrac|Θ|360°}}) proportional part of the whole circumference (see arc). So:

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

Image:Angle central reflex.svg

{{math proof|title=Proof (for radians)|proof= The circumference of a circle with radius {{mvar|R}} is {{math|2πR}}, and the minor arc {{math|L}} is the ({{sfrac|Θ|2π}}) proportional part of the whole circumference (see arc). So

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

If the central angle {{math|Θ}} is not subtended by the minor arc {{math|L}}, then {{math|Θ}} is a reflex angle and

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

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 (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 2\pi/n.

See also

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

Angle