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

$A=\pi k^2$ (1)

for finding the area of a spherical calotte.

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