calculating the area on the surface os a sphere created by intersection of two spherical caps!

Question

Consider a spherical object composed of two compartments (A and B, not necessarily hemispheres) sitting at the interface which is characterized by a plane separating 1 and 2. For this case, calculating the various surface area of contact (namely A1, A2, B1, B2) is simple using the formula for spherical caps. My question is about the case that happens after rotating the particle through the plane of interface! I am trying to figure out the surface area of the contact between different phases. In fact, I am interested in the two unequal areas on the surface of the sphere formed by the cross section of two planes with the intersection line of length less than the sphere diameter(if their intersection was passing through the center with length equal to the diameter, the answer was simply the area of the lune!!! But this is not what I am looking for, my case is more general). I also found a similar question on this website but could not relate it to my case. Here is the link: Calculate the area on a sphere of the intersection of two spherical caps. Thank