Circular sector

<h2 id="types">Types</h2>
<p>A sector with the central angle of 180° is called a <i><a href="/facts/Disk_(geometry)/DgMVr97H">half-disk</a></i> and is bounded by a <a href="/facts/Diameter/6uNn4VmU">diameter</a> and a <a href="/facts/Semicircle/Im66xpfu">semicircle</a>. Sectors with other central angles are sometimes given special names, such as quadrants (90°), sextants (60°), and octants (45°), which come from the sector being one quarter, sixth or eighth part of a full circle, respectively. The <a href="/facts/Arc_(geometry)/xBQ0pTBW">arc</a> of a quadrant (a <a href="/facts/Circular_arc/RrehhCQR">circular arc</a>) can also be termed a quadrant.
</p>
<h2 id="area">Area</h2>
<p class="note">See also: <a href="/facts/Circular_arc/RrehhCQR">Circular arc § Sector area</a></p>
<p>The total area of a circle is <i>πr</i>2. The area of the sector can be obtained by multiplying the circle's area by the ratio of the angle θ (expressed in radians) and 2<i>π</i> (because the area of the sector is directly proportional to its angle, and 2<i>π</i> is the angle for the whole circle, in radians):

A
        =
        π
        
          r
          
            2
          
        
        
        
          
            θ
            
              2
              π
            
          
        
        =
        
          
            
              
                r
                
                  2
                
              
              θ
            
            2
          
        
      
    
    {\displaystyle A=\pi r^{2}\,{\frac {\theta }{2\pi }}={\frac {r^{2}\theta }{2}}}

</p><p>The area of a sector in terms of L can be obtained by multiplying the total area <i>πr</i>2 by the ratio of L to the total perimeter 2<i>πr</i>.

A
        =
        π
        
          r
          
            2
          
        
        
        
          
            L
            
              2
              π
              r
            
          
        
        =
        
          
            
              r
              L
            
            2
          
        
      
    
    {\displaystyle A=\pi r^{2}\,{\frac {L}{2\pi r}}={\frac {rL}{2}}}

</p><p>Another approach is to consider this area as the result of the following integral:

A
        =
        
          ∫
          
            0
          
          
            θ
          
        
        
          ∫
          
            0
          
          
            r
          
        
        d
        S
        =
        
          ∫
          
            0
          
          
            θ
          
        
        
          ∫
          
            0
          
          
            r
          
        
        
          
            
              r
              ~
            
          
        
        
        d
        
          
            
              r
              ~
            
          
        
        
        d
        
          
            
              θ
              ~
            
          
        
        =
        
          ∫
          
            0
          
          
            θ
          
        
        
          
            1
            2
          
        
        
          r
          
            2
          
        
        
        d
        
          
            
              θ
              ~
            
          
        
        =
        
          
            
              
                r
                
                  2
                
              
              θ
            
            2
          
        
      
    
    {\displaystyle A=\int _{0}^{\theta }\int _{0}^{r}dS=\int _{0}^{\theta }\int _{0}^{r}{\tilde {r}}\,d{\tilde {r}}\,d{\tilde {\theta }}=\int _{0}^{\theta }{\frac {1}{2}}r^{2}\,d{\tilde {\theta }}={\frac {r^{2}\theta }{2}}}

</p><p>Converting the central angle into <a href="/facts/Degree_(angle)/f48Ns8SX">degrees</a> gives<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>

A
        =
        π
        
          r
          
            2
          
        
        
          
            
              θ
              
                ∘
              
            
            
              360
              
                ∘
              
            
          
        
      
    
    {\displaystyle A=\pi r^{2}{\frac {\theta ^{\circ }}{360^{\circ }}}}

</p>
<h2 id="perimeter">Perimeter</h2>
<p>The length of the <a href="/facts/Perimeter/yQv1AajA">perimeter</a> of a sector is the sum of the arc length and the two radii:

P
        =
        L
        +
        2
        r
        =
        θ
        r
        +
        2
        r
        =
        r
        (
        θ
        +
        2
        )
      
    
    {\displaystyle P=L+2r=\theta r+2r=r(\theta +2)}

where θ is in radians.
</p>
<h2 id="arc-length">Arc length</h2>
<p>The formula for the length of an arc is:<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>

L
        =
        r
        θ
      
    
    {\displaystyle L=r\theta }

where L represents the arc length, r represents the radius of the circle and θ represents the angle in radians made by the arc at the centre of the circle.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p><p>If the value of angle is given in degrees, then we can also use the following formula by:<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>

L
        =
        2
        π
        r
        
          
            θ
            360
          
        
      
    
    {\displaystyle L=2\pi r{\frac {\theta }{360}}}

</p>
<h2 id="chord-length">Chord length</h2>
<p>The length of a <a href="/facts/Chord_(geometry)/aAyoEdLd">chord</a> formed with the extremal points of the arc is given by

C
        =
        2
        R
        sin
        ⁡
        
          
            θ
            2
          
        
      
    
    {\displaystyle C=2R\sin {\frac {\theta }{2}}}

where C represents the chord length, R represents the radius of the circle, and θ represents the angular width of the sector in radians.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Circular_segment/mkr8SBoq">Circular segment</a> – the part of the sector which remains after removing the triangle formed by the center of the circle and the two endpoints of the circular arc on the boundary.</li>
<li><a href="/facts/Conic_section/teqgLptX">Conic section</a></li>
<li><a href="/facts/Earth_quadrant/Ic30hDn1">Earth quadrant</a></li>
<li><a href="/facts/Hyperbolic_sector/PCEFy8fd">Hyperbolic sector</a></li>
<li><a href="/facts/Sector_of_(mathematics)/1JA8Tsgb">Sector of (mathematics)</a></li>
<li><a href="/facts/Spherical_sector/oPrRNJnI">Spherical sector</a> – the analogous 3D figure</li></ul>

<h2 id="sources">Sources</h2>
<ul><li>Gerard, L. J. V. (1874). <a href="https://archive.org/details/elementsgeometr00geragoog/page/n309"><i>The Elements of Geometry, in Eight Books; or, First Step in Applied Logic</i></a>. London: <a href="/facts/Longman/l5JQ1NGz">Longmans, Green, Reader and Dyer</a>. p. 285.</li>
<li><a href="/facts/Adrien-Marie_Legendre/2a7CAolM">Legendre, Adrien-Marie</a> (1858). <a href="/facts/Charles_Davies_(professor)/oIGe2P2T">Davies, Charles</a> (ed.). <a href="https://books.google.com/books?id=pFYliSRwxEgC&pg=RA1-PA119"><i>Elements of Geometry and Trigonometry</i></a>. New York: <a href="/facts/Alfred_Smith_Barnes/QFW0gSDe">A. S. Barnes & Co.</a> p. 119.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Dewan, Rajesh K. (2016). Saraswati Mathematics. New Delhi: New Saraswati House India Pvt Ltd. p. 234. ISBN 978-8173358371. <a href="978-8173358371" target="_blank">978-8173358371</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Achatz, Thomas; Anderson, John G. (2005). Technical shop mathematics. Kathleen McKenzie (3rd ed.). New York: Industrial Press. p. 376. ISBN 978-0831130862. OCLC 56559272. <a href="978-0831130862" target="_blank">978-0831130862</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Uppal, Shveta (2019). Mathematics: Textbook for class X. New Delhi: National Council of Educational Research and Training. pp. 226, 227. ISBN 978-81-7450-634-4. OCLC 1145113954. <a href="978-81-7450-634-4" target="_blank">978-81-7450-634-4</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Larson, Ron; Edwards, Bruce H. (2002). Calculus I with Precalculus (3rd ed.). Boston, MA.: Brooks/Cole. p. 570. ISBN 978-0-8400-6833-0. OCLC 706621772. <a href="978-0-8400-6833-0" target="_blank">978-0-8400-6833-0</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Wicks, Alan (2004). Mathematics Standard Level for the International Baccalaureate : a text for the new syllabus. West Conshohocken, PA: Infinity Publishing.com. p. 79. ISBN 0-7414-2141-0. OCLC 58869667. <a href="0-7414-2141-0" target="_blank">0-7414-2141-0</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Uppal (2019). - Uppal, Shveta (2019). Mathematics: Textbook for class X. New Delhi: National Council of Educational Research and Training. pp. 226, 227. ISBN 978-81-7450-634-4. OCLC 1145113954. <a href="http://www.ncert.nic.in/ncerts/l/jemh112.pdf#page=4" target="_blank">http://www.ncert.nic.in/ncerts/l/jemh112.pdf#page=4</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
</ol>

Circular sector open-in-new

Circular sector