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\begin{figure}\begin{center}\BoxedEPSF{Sagitta.epsf scaled 1400}\end{center}\end{figure}

The Perpendicular distance $s$ from an Arc's Midpoint to the Chord across it, equal to the Radius $r$ minus the Apothem $a$,

\end{displaymath} (1)

For a regular Polygon of side length $a$,
$\displaystyle s$ $\textstyle \equiv$ $\displaystyle R-r= {\textstyle{1\over 2}}a\left[{\csc\left({\pi\over n}\right)-\cot\left({\pi\over n}\right)}\right]$  
  $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}a\tan\left({\pi\over 2n}\right)$ (2)
  $\textstyle =$ $\displaystyle r\tan\left({\pi\over n}\right)\tan\left({\pi\over 2n}\right)$ (3)
  $\textstyle =$ $\displaystyle 2R\sin^2\left({\pi\over 2n}\right),$ (4)

where $R$ is the Circumradius, $r$ the Inradius, $a$ is the side length, and $n$ is the number of sides.

See also Apothem, Chord, Sector, Segment

© 1996-9 Eric W. Weisstein