area of spherical calotte by means of chordAreaOfSphericalCalotteByMeansOfChord2008-08-18 12:42:222008-08-18 12:44:13Derivationarea of spherical zonespherical calotte% this is the default PlanetMath preamble. as your knowledge
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\PMlinkescapeword{height}
Let the arc $PR$ of a circle with radius $r$ rotate about the diameter $PQ$.\, The surface of revolution is a spherical calotte with the height $h$.\, If the \PMlinkescapetext{length} of the chord $PR$ is $k$, we obtain from the right triangle $PQR$ the proportion equation
$$\frac{h}{k} = \frac{k}{2r},$$
i.e. the chord $k$ is the central proportional of the height and the diameter.\, Accordingly, we can substitute\, $2rh = k^2$\, to the expression
$$A = 2\pi rh$$
of the area of the spherical calotte derived in the \PMlinkname{parent entry}{AreaOfSphericalZone}. Thus we have an alternative \PMlinkescapetext{formula}
\begin{align}
A = \pi{k}^2
\end{align}
for finding the area of a spherical calotte.
\begin{thebibliography}{9}
\bibitem{K.V.} {\sc K. V\"ais\"al\"a}: {\em Geometria}.\, Kymmenennen painoksen muuttamaton lis\"apainos.\, Werner S\"oderstr\"om Osakeyhti\"o, Porvoo \& Helsinki (1971).
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