TLDR;
What's the probability law of the area of a random cut on a sphere, with the random cut being perpendicular to a fixed segment connecting two antipodal points on the sphere, and sampled uniformly?
More precise description and my results
Suppose to have a sphere of radius 1, to choose a segment connecting two antipodal points and to randomly sample (with uniform distribution) a point on this segment. Now draw the 2d plane orthogonal to the segment and passing by the point.
This plan will cut the sphere in two, determining a circle. I would like to find the distribution of the areas of such circles.
Using the transfer operator, unless I did some trivial algebra errors, I have showed that called $P(A)$ the random variable of the area of such circle, we have that (the support is $[0, \pi]$), $P(A)$ has a density, and it is
$$ \frac{1}{2\pi\sqrt{(1 - A/\pi)}}. $$
Since I need this result for a computational problem I would be fine with this, but I wonder it this density is coming from a know probability law. Any help is appreciated :)
As requested in comments:
It is a stretched (by a factor of $\pi$) version of a $\text{Beta}\left(1,\frac12\right)$ distribution
See Wikipedia on the Beta distribution for some of the statistics, remembering to scale where necessary