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Difference between revisions of "Sector"

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<div style="float:right"><asy>size(150);
 
<div style="float:right"><asy>size(150);
−
real angle1=30, angle2=100;
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real angle1=30, angle2=120;
 
pair O=origin, A=dir(angle2), B=dir(angle1);
 
pair O=origin, A=dir(angle2), B=dir(angle1);
 
path sector=O--B--arc(O,1,angle1,angle2)--A--cycle;
 
path sector=O--B--arc(O,1,angle1,angle2)--A--cycle;
Line 7: Line 7:
 
D(A--O--B);
 
D(A--O--B);
 
MP("O",D(O),SSW);
 
MP("O",D(O),SSW);
−
MP("A",D(A),NNW);
+
MP("A",D(A),NW);
 
MP("B",D(B),NE);
 
MP("B",D(B),NE);
−
MP("\theta",(0.075,0.075),N);</asy></div>
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MP("\theta",(0.05,0.075),N);</asy></div>
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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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A '''sector''' of a [[circle]] is a region bounded by two [[radius|radii]] of the circle and an [[arc]].
  
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== Area ==
+
If the [[central angle]] of the sector is <math>\pi</math> (or <math>180^{\circ}</math>), then the sector is a [[semicircle]].
−
The [[area]] of a sector <math>AOB</math> 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]], <math>\angle AOB</math>, in radians.
 
  
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Therefore, the area of a sector <math>AOB</math> is <math>\frac{r^2\theta}{2}</math>, where <math>r</math> is the radius and <math>\theta=\angle AOB</math> is in radians.
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==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>.
 
Alternatively, if <math>\theta</math> is in degrees, the area is <math>\frac{\pi r^2\theta}{360^{\circ}}</math>.

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]

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 $\pi$ (or $180^{\circ}$), then the sector is a semicircle.

Area

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

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

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

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