# area of spherical calotte by means of chord

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 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 (http://planetmath.org/AreaOfSphericalZone). Thus we have an alternative

 $\displaystyle A=\pi{k}^{2}$ (1)

for finding the area of a spherical calotte.

## References

• 1 K. Väisälä: Geometria.  Kymmenennen painoksen muuttamaton lisäpainos.  Werner Söderström Osakeyhtiö, Porvoo & Helsinki (1971).
Title area of spherical calotte by means of chord AreaOfSphericalCalotteByMeansOfChord 2013-03-22 18:19:20 2013-03-22 18:19:20 pahio (2872) pahio (2872) 5 pahio (2872) Derivation msc 51M04 alternative way to find area of spherical calotte ThalesTheorem SimilarityOfTriangles