area of spherical calotte by means of chord

Let the arc PR of a circle with radius r rotate about the diameterMathworldPlanetmath PQ.  The surface of revolutionMathworldPlanetmath is a spherical calotte with the height h.  If the of the chord PR is k, we obtain from the right triangleMathworldPlanetmath PQR the proportion equation


i.e. the chord k is the central proportional of the height and the diameter.  Accordingly, we can substitute  2rh=k2  to the expression


of the area of the spherical calotte derived in the parent entry ( Thus we have an alternative

A=πk2 (1)

for finding the area of a spherical calotte.


  • 1 K. Väisälä: Geometria.  Kymmenennen painoksen muuttamaton lisäpainos.  Werner Söderström Osakeyhtiö, Porvoo & Helsinki (1971).
Title area of spherical calotte by means of chord
Canonical name AreaOfSphericalCalotteByMeansOfChord
Date of creation 2013-03-22 18:19:20
Last modified on 2013-03-22 18:19:20
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 5
Author pahio (2872)
Entry type Derivation
Classification msc 51M04
Synonym alternative way to find area of spherical calotte
Related topic ThalesTheorem
Related topic SimilarityOfTriangles