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.