Calculate the area on a sphere of the intersection of two spherical caps Given a sphere of radius $r$ with two spherical caps on it defined by the radii ($a_1$ and $a_2$) of the bases of the spherical caps, given a separation of the two spherical caps by angle $\theta$, how do you calculate the surface area of that intersection? 
To clarify, the area is that of the curved surface on the sphere defined by the intersection. At the extreme where both $a_1,a_2 = r$, we would be describing a spherical lune. 
Alternatively define the spherical caps by the angles 
$\Phi_1 = \arcsin(a_1/r)$ and $\Phi_2 = \arcsin(a_2/r)$.

Here's a simplified formula as a function of your 3 variables, $a_1$, $a_2$, and $\theta$:
$$
2\cos(a_2)\arccos \left ( \frac{-\cos(a_1) + \cos(\theta)\cos(a_2)}{\sin(\theta)\sin(a_2)} \right ) \\
-2\cos(a_1)\arccos \left ( \frac{\cos(a_2) - \cos(\theta)\cos(a_1)}{\sin(\theta)\sin(a_1)} \right ) \\
-2\arccos \left ( \frac{-\cos(\theta) + \cos(a_1)\cos(a_2)}{\sin(a_1)\sin(a_2)} \right ) \\
-2\pi\cos(a_2)
$$
As previously stated, the caps must intersect and one cannot entirely contain the other.
This solution is copied from a graphics presentation by AMD, and is originally from a biochemistry paper. I have no proof that it works, so take it for what it's worth.
TOVCHIGRECHKO, A. AND VAKSER, I.A. 2001. How
common is the funnel-like energy landscape in protein-protein
interactions? Protein Sci. 10:1572-1583
