Difference between revisions of "Sector"

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MP("O",D(O),SSW);
 
MP("O",D(O),SSW);
 
MP("A",D(A),NNW);
 
MP("A",D(A),NNW);
MP("B",D(B),NE);</asy></div>
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MP("B",D(B),NE);
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MP("\theta",(0.075,0.075),N);</asy></div>
 
A '''sector''' of a [[circle]] <math>O</math> is a region bounded by two [[radius|radii]] of the circle, <math>OA</math> and <math>OB</math>, and the [[arc]] <math>AB</math>.
 
A '''sector''' of a [[circle]] <math>O</math> is a region bounded by two [[radius|radii]] of the circle, <math>OA</math> and <math>OB</math>, and the [[arc]] <math>AB</math>.
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== Area ==
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The [[area]] of a sector <math>AOB</math>, where <math>\theta=\angle AOB</math> is in radians, is found by [[multiply]]ing the area of circle <math>O</math> by <math>\frac{\theta}{2\pi}</math>.
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Therefore, the area of a sector <math>AOB</math>, where <math>r</math> is the radius and <math>\theta=\angle AOB</math> is in radians, is <math>\frac{\theta r^2}{2}</math>.
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Alternatively, if <math>\theta</math> is in degrees, the area is <math>\frac{\theta \pi r^2}{360^{\circ}}</math>.
  
 
{{stub}}
 
{{stub}}
 
[[Category:Definition]]
 
[[Category:Definition]]
 
[[Category:Geometry]]
 
[[Category:Geometry]]

Revision as of 20:37, 24 April 2008

[asy]size(150); real angle1=30, angle2=100; pair O=origin, A=dir(angle2), B=dir(angle1); path sector=O--B--arc(O,1,angle1,angle2)--A--cycle; fill(sector,gray(0.9)); D(unitcircle); D(A--O--B); MP("O",D(O),SSW); MP("A",D(A),NNW); MP("B",D(B),NE); MP("\theta",(0.075,0.075),N);[/asy]

A sector of a circle $O$ is a region bounded by two radii of the circle, $OA$ and $OB$, and the arc $AB$.

Area

The area of a sector $AOB$, where $\theta=\angle AOB$ is in radians, is found by multiplying the area of circle $O$ by $\frac{\theta}{2\pi}$.

Therefore, the area of a sector $AOB$, where $r$ is the radius and $\theta=\angle AOB$ is in radians, is $\frac{\theta r^2}{2}$.

Alternatively, if $\theta$ is in degrees, the area is $\frac{\theta \pi r^2}{360^{\circ}}$.

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