Intuitive explanation for $Area_{\text{sphere}}\times\ Area_{\text{cross-section}}=Area_{\text{major cap}}\times Area_{\text{minor cap}}$

Question

A sphere and a plane intersect to produce a circular intersection and two spherical caps.

$S=$ surface area of sphere
$C=$ area of circular intersection
$A=$ surface area of major cap
$B=$ surface area of minor cap

I have discovered that:

$$S\times C=A\times B$$

Proof:

Let $r=$ radius of sphere, $h=$ height of major cap.
Using the formula for the surface area of a spherical cap,
$A\times B=2\pi rh\times 2\pi r(2r-h)=4\pi r^2\times \pi(r^2-(r-h)^2)=S\times C$

The simplicity of the result makes me wonder if there is an intuitive explanation. (Here is an example of an intuitive explanation of a geometric result.)

So my question is:

Is there an intuitive explanation for the fact that $S\times C=A\times B$ ?