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\begin{figure}\begin{center}\BoxedEPSF{Segment.epsf scaled 1000}\end{center}\end{figure}

A portion of a Circle whose upper boundary is a circular Arc and whose lower boundary is a Chord making a Central Angle $\theta<\pi$ radians (180°), illustrated above as the shaded region. Let $R$ be the radius of the Circle, $c$ the Chord length, $s$ the Arc Length, $h$ the height of the arced portion, and $d$ the height of the triangular portion. Then

$\displaystyle R$ $\textstyle =$ $\displaystyle h+d$ (1)
$\displaystyle s$ $\textstyle =$ $\displaystyle R\theta$ (2)
$\displaystyle d$ $\textstyle =$ $\displaystyle R\cos({\textstyle{1\over 2}}\theta)$ (3)
  $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}c\cot({\textstyle{1\over 2}}\theta)$ (4)
  $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\sqrt{4R^2-c^2}$ (5)
$\displaystyle c$ $\textstyle =$ $\displaystyle 2R\sin({\textstyle{1\over 2}}\theta)$ (6)
  $\textstyle =$ $\displaystyle 2d\tan({\textstyle{1\over 2}}\theta)$ (7)
  $\textstyle =$ $\displaystyle 2\sqrt{R^2-d^2}$ (8)
  $\textstyle =$ $\displaystyle 2\sqrt{h(2R-h)}\,.$ (9)

The Angle $\theta$ obeys the relationships
$\displaystyle \theta$ $\textstyle =$ $\displaystyle {s\over R}=2\cos^{-1}\left({d\over R}\right)=2\tan^{-1}\left({c\over 2d}\right)$  
  $\textstyle =$ $\displaystyle 2\sin^{-1}\left({c\over 2R}\right).$ (10)

The Area of the segment is then
$\displaystyle A$ $\textstyle =$ $\displaystyle A_{\rm sector}-A_{\rm isosceles\ triangle}$ (11)
  $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}R^2(\theta-\sin\theta)$ (12)
  $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}(Rs-cd)$ (13)
  $\textstyle =$ $\displaystyle R^2\cos^{-1}\left({d\over R}\right)-d\sqrt{R^2-d^2}$ (14)
  $\textstyle =$ $\displaystyle R^2\cos^{-1}\left({R-h\over R}\right)-(R-h)\sqrt{2Rh-h^2},$ (15)

where the formula for the Isosceles Triangle in terms of the Vertex angle has been used (Beyer 1987).

See also Chord, Circle-Circle Intersection, Cylindrical Segment, Lens, Parabolic Segment, Sagitta, Sector, Spherical Segment


Beyer, W. H. (Ed.) CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 125, 1987.

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© 1996-9 Eric W. Weisstein