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
hk=k2r, |
i.e. the chord k is the central proportional of the height and the diameter. Accordingly, we can substitute 2rh=k2 to the expression
A=2πrh |
of the area of the spherical calotte derived in the parent entry (http://planetmath.org/AreaOfSphericalZone). Thus we have an alternative
A=πk2 | (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 |
---|---|
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 |