Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
I would say a picture might it make more clear as to what you are trying to calculate. Or at least consider re-phrasing the question. What is 1 and 2 in your question? – Maesumi Jan 10 '13 at 20:45
Image upload is not possible for new users! Suppose that a particle is sitting at the interface of two liquids labeled as (1) and (2) separated by plane of interface. The particle itself has two compartments (A) and (B) separated by a plane (particle plane). The contact area of the particle's components A, B with each liquid (1), (2) is of interest. When the plane of interface is parallel to the particle plane using the spherical cap formula one can calculate areas! what happens when these two planes intersect? (the intersection line has a length less than the particle diameter). Thanks – Sepideh Jan 10 '13 at 21:15
So is this the right rephrasing: Two planes intersect a sphere and divide it into 4 wedges. What is the spherical surface area of each wedge? – Maesumi Jan 10 '13 at 21:27
If so then your problem is a case of spherical triangle even though there are only two sides to this triangle. There are a variety of formulas for finding its area. – Maesumi Jan 11 '13 at 0:53
yes you are right! this is exactly the problem I am trying to solve ! whatever I have seen so far on the spherical triangle is the shape which is formed by the intersection of three great circles, How can I relate this to my problem? (as you correctly mentioned, the intersection of two circles which are not the great circles). Would you direct me to some references? – Sepideh Jan 14 '13 at 14:30

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.