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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.

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

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