Chord (geometry)

<h2 id="in-circles">In circles</h2>
<p class="note">Main article: <a href="/facts/Circle/9fy5ggKn">Circle § Chord</a></p>
<p>Among properties of chords of a <a href="/facts/Circle/9fy5ggKn">circle</a> are the following:
</p>
<ol><li>Chords are equidistant from the center if and only if their lengths are equal.</li>
<li>Equal chords are subtended by equal angles from the center of the circle.</li>
<li>A chord that passes through the center of a circle is called a diameter and is the longest chord of that specific circle.</li>
<li>If the line extensions (secant lines) of chords AB and CD intersect at a point P, then their lengths satisfy AP·PB = CP·PD (<a href="/facts/Power_of_a_point_theorem/KVt6VnD6">power of a point theorem</a>).</li></ol>
<h2 id="in-conics">In conics</h2>
<p>The midpoints of a set of parallel chords of a <a href="/facts/Conic/teqgLptX">conic</a> are <a href="/facts/Collinearity/SxbVBJYR">collinear</a> (<a href="/facts/Midpoint_theorem_(conics)/9zDDrrmd">midpoint theorem for conics</a>).<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p>
<h2 id="in-trigonometry">In trigonometry</h2>

<p>Chords were used extensively in the early development of <a href="/facts/Trigonometry/K4hfaC6F">trigonometry</a>. The first known trigonometric table, compiled by <a href="/facts/Hipparchus/pHld5PfE">Hipparchus</a> in the 2nd century BC, is no longer extant but <a href="/facts/Table_of_chords/h6Ubj0sq">tabulated the value of the chord function</a> for every ⁠7+1/2⁠ <a href="/facts/Degree_(angle)/f48Ns8SX">degrees</a>.  In the 2nd century AD, <a href="/facts/Ptolemy/htudGQTQ">Ptolemy</a> compiled a more extensive table of chords in <a href="/facts/Almagest/30qaBRAq">his book on astronomy</a>, giving the value of the chord for angles ranging from ⁠1/2⁠ to 180 degrees by increments of ⁠1/2⁠ degree. Ptolemy used a circle of diameter 120, and gave chord lengths accurate to two <a href="/facts/Sexagesimal/bspNRWdR">sexagesimal</a> (base sixty) digits after the integer part.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a>
</p><p>The chord function is defined geometrically as shown in the picture. The chord of an <a href="/facts/Angle/gFUTEQYq">angle</a> is the <a href="/facts/Length/EwFyAqGH">length</a> of the chord between two points on a unit circle separated by that <a href="/facts/Central_angle/h00sYLNj">central angle</a>. The angle <i>θ</i> is taken in the positive sense and must lie in the interval 0 < <i>θ</i> ≤ π (radian measure). The chord function can be related to the modern <a href="/facts/Sine/gQ6LXK8z">sine</a> function, by taking one of the points to be (1,0), and the other point to be (cos <i>θ</i>, sin <i>θ</i>), and then using the <a href="/facts/Pythagorean_theorem/Lh6x2Kr5">Pythagorean theorem</a> to calculate the chord length:<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>

crd
        ⁡
        θ
        =
        
          
            (
            1
            −
            cos
            ⁡
            θ
            
              )
              
                2
              
            
            +
            
              sin
              
                2
              
            
            ⁡
            θ
          
        
        =
        
          
            2
            −
            2
            cos
            ⁡
            θ
          
        
        =
        2
        sin
        ⁡
        
          (
          
            
              θ
              2
            
          
          )
        
        .
      
    
    {\displaystyle \operatorname {crd} \theta ={\sqrt {(1-\cos \theta )^{2}+\sin ^{2}\theta }}={\sqrt {2-2\cos \theta }}=2\sin \left({\frac {\theta }{2}}\right).}
  
 <a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
<p>The last step uses the <a href="/facts/List_of_trigonometric_identities/Xw9Jx8Vz">half-angle formula</a>. Much as modern trigonometry is built on the sine function, ancient trigonometry was built on the chord function. Hipparchus is purported to have written a twelve-volume work on chords, all now lost, so presumably, a great deal was known about them. In the table below (where <i>c</i> is the chord length, and <i>D</i> the diameter of the circle) the chord function can be shown to satisfy many identities analogous to well-known modern ones:
</p>
<table><tbody><tr><th>Name</th><th>Sine-based</th><th>Chord-based</th></tr><tr><td>Pythagorean</td><td>                              sin                      2                          ⁡        θ        +                  cos                      2                          ⁡        θ        =        1                      {\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1\,}  </td><td>                              crd                      2                          ⁡        θ        +                  crd                      2                          ⁡        (        π        −        θ        )        =        4                      {\displaystyle \operatorname {crd} ^{2}\theta +\operatorname {crd} ^{2}(\pi -\theta )=4\,}  </td></tr><tr><td>Half-angle</td><td>                    sin        ⁡                              θ            2                          =        ±                                                            1                −                cos                ⁡                θ                            2                                                    {\displaystyle \sin {\frac {\theta }{2}}=\pm {\sqrt {\frac {1-\cos \theta }{2}}}\,}  </td><td>                    crd        ⁡                                       θ            2                          =                              2            −            crd            ⁡            (            π            −            θ            )                                        {\displaystyle \operatorname {crd} \ {\frac {\theta }{2}}={\sqrt {2-\operatorname {crd} (\pi -\theta )}}\,}  </td></tr><tr><td><a href="/facts/Apothem/CHbBbI8t">Apothem</a> (<i>a</i>)</td><td>                    c        =        2                                            r                              2                                      −                          a                              2                                                          {\displaystyle c=2{\sqrt {r^{2}-a^{2}}}}  </td><td>                    c        =                                            D                              2                                      −            4                          a                              2                                                          {\displaystyle c={\sqrt {D^{2}-4a^{2}}}}  </td></tr><tr><td>Angle (<i>θ</i>)</td><td>                    c        =        2        r        sin        ⁡                  (                                    θ              2                                )                      {\displaystyle c=2r\sin \left({\frac {\theta }{2}}\right)}  </td><td>                    c        =                              D            2                          crd        ⁡        θ              {\displaystyle c={\frac {D}{2}}\operatorname {crd} \theta }  </td></tr></tbody></table>
<p>The inverse function exists as well:<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p>

θ
        =
        2
        arcsin
        ⁡
        
          
            c
            
              2
              r
            
          
        
      
    
    {\displaystyle \theta =2\arcsin {\frac {c}{2r}}}

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Circular_segment/mkr8SBoq">Circular segment</a> - the part of the sector that 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/Scale_of_chords/hSJhVkUH">Scale of chords</a></li>
<li><a href="/facts/Ptolemy%2527s_table_of_chords/h6Ubj0sq">Ptolemy's table of chords</a></li>
<li><a href="/facts/Holditch%2527s_theorem/x9TKB2YB">Holditch's theorem</a>, for a chord rotating in a convex closed curve</li>
<li><a href="/facts/Circle_graph/mI27LMXp">Circle graph</a></li>
<li><a href="/facts/Exsecant_and_excosecant/CNKrYMjC">Exsecant and excosecant</a></li>
<li><a href="/facts/Versine_and_haversine/MllE4ISd">Versine and haversine</a> - (
  
    
      
        crd
        ⁡
        θ
        =
        2
        
          
            haversin
            ⁡
            θ
          
        
      
    
    {\displaystyle \operatorname {crd} \theta =2{\sqrt {\operatorname {haversin} \theta }}}
  
)</li>
<li><a href="/facts/Zindler_curve/317CIeeh">Zindler curve</a> (closed and simple curve in which all chords that divide the arc length into halves have the same length)</li>
<li><a href="/facts/Bertrand_paradox_(probability)/3rFRzsfg">Bertrand paradox (probability)</a> average length of chord paradox</li></ul>

<h2 id="external-links">External links</h2>

Wikimedia Commons has media related to Chord (geometry).

<ul><li><a href="http://aleph0.clarku.edu/~djoyce/ma105/trighist.html">History of Trigonometry Outline</a></li>
<li><a href="http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Trigonometric_functions.html">Trigonometric functions</a> <a href="https://web.archive.org/web/20170310000525/http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Trigonometric_functions.html">Archived</a> 2017-03-10 at the <a href="/facts/Wayback_Machine/nmQ3a6JC">Wayback Machine</a>, focusing on history</li>
<li><a href="http://www.mathopenref.com/chord.html">Chord (of a circle)</a> With interactive animation</li></ul>

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

<ol>
<li id="fn:1"><p>Gibson, C. G. (2003). "7.1 Midpoint Loci". Elementary Euclidean Geometry: An Introduction. Cambridge University Press. pp. 65–68. ISBN 9780521834483. <a href="9780521834483" target="_blank">9780521834483</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Maor, Eli (1998). Trigonometric Delights. Princeton University Press. pp. 25–27. ISBN 978-0-691-15820-4. <a href="978-0-691-15820-4" target="_blank">978-0-691-15820-4</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Maor, Eli (1998). Trigonometric Delights. Princeton University Press. pp. 25–27. ISBN 978-0-691-15820-4. <a href="978-0-691-15820-4" target="_blank">978-0-691-15820-4</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Weisstein, Eric W. "Circular Segment". From MathWorld--A Wolfram Web Resource. <a href="https://mathworld.wolfram.com/CircularSegment.html" target="_blank">https://mathworld.wolfram.com/CircularSegment.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Simpson, David G. (2001-11-08). "AUXTRIG" (FORTRAN-90 source code). Greenbelt, Maryland, US: NASA Goddard Space Flight Center. Retrieved 2015-10-26. <a href="http://www.davidgsimpson.com/software/auxtrig_f90.txt" target="_blank">http://www.davidgsimpson.com/software/auxtrig_f90.txt</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
</ol>

Chord (geometry) open-in-new

Chord (geometry)