# disk-disk intersection area

I have two disks of radii $R_1, R_2$ with distance between centers, $d < R_1 + R_2$.

How can I find the surface area common to the two disks?

Rationale:

Solar irradiation / energy input in penumbra during solar eclipse, a problem for sun-synchronous satellites. Knowing apparent size of the Sun and the Moon, and position within penumbra it's possible to calculate how much of the solar disk remains unobscured, as a simple difference between its surface and the intersection area.

• Circles or spheres? circles don't have common Surface areas. – Win Vineeth Feb 8 '16 at 13:35
• @WinVineeth: Ok, edited. Disks. Sun's irradiation is approximately constant over its apparent area. – SF. Feb 8 '16 at 13:43
• this should get you started math.stackexchange.com/questions/402858/… – Charlie Feb 8 '16 at 13:56
• @Charlie: that one is for $R_1=R_2=d$. I'll see what I can do with it. – SF. Feb 8 '16 at 14:04

The intersection is a composite of two circular segments. You can calculate the areas of these circular segments after finding the angles in the triangle with sides $R1$, $R2$ and $d$.