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 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 parent entry. Thus we have an alternative formula

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Bibliography

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K. V¨AISÄLÄ: Geometria. Kymmenennen painoksen muuttamaton lisäpainos. Werner Söderström Osakeyhtiö, Porvoo & Helsinki (1971).