Difference between revisions of "Sector"
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| − | A '''sector''' of a [[circle]] | + | <div style="float:right"><asy>size(150); |
| + | real angle1=30, angle2=120; | ||
| + | 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),NW); | ||
| + | MP("B",D(B),NE); | ||
| + | MP("\theta",(0.05,0.075),N);</asy></div> | ||
| + | A '''sector''' of a [[circle]] is a region bounded by two [[radius|radii]] of the circle and an [[arc]]. | ||
| + | If the [[central angle]] of the sector is <math>\pi</math> (or <math>180^{\circ}</math>), then the sector is a [[semicircle]]. | ||
| + | |||
| + | ==Area== | ||
| + | The [[area]] of a sector is found by [[multiply]]ing the area of circle <math>O</math> by <math>\frac{\theta}{2\pi}</math>, where <math>\theta</math> is the central angle in radians. | ||
| + | |||
| + | Therefore, the area of a sector is <math>\frac{r^2\theta}{2}</math>, where <math>r</math> is the radius and <math>\theta</math> is the central angle in radians. | ||
| + | |||
| + | Alternatively, if <math>\theta</math> is in degrees, the area is <math>\frac{\pi r^2\theta}{360^{\circ}}</math>. | ||
{{stub}} | {{stub}} | ||
| + | [[Category:Definition]] | ||
| + | [[Category:Geometry]] | ||
Latest revision as of 20:12, 24 April 2008
![[asy]size(150); real angle1=30, angle2=120; 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),NW); MP("B",D(B),NE); MP("\theta",(0.05,0.075),N);[/asy]](http://latex.artofproblemsolving.com/7/c/9/7c9a7756f90fe421d7f60125f66669fa9200d25d.png)
A sector of a circle is a region bounded by two radii of the circle and an arc.
If the central angle of the sector is 
(or 
), then the sector is a semicircle.
Area
The area of a sector is found by multiplying the area of circle 
by 
, where 
is the central angle in radians.
Therefore, the area of a sector is 
, where 
is the radius and is the central angle in radians.
Alternatively, if is in degrees, the area is 
.
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