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How to determine the area of intersecting circles?

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Oct27-13, 12:44 AM
P: 344
I've been wondering how to calculate the area of intersection of two overlapping circles in terms of their radii. There's two cases I'm interested in:

The easier case:
Suppose there are two circles of radius R and r (R > r). The center of the larger circle is at the origin, and the center of the other circle is at (x,y)=(R,0). How can I find the area of the overlap in terms of R and r? I can't think of any clear way to do this by hand.

The more complicated case:
Do the same except let the center of the smaller circle lie between (R-r,0) < (x,y) < (R+r,0)

I've attached a drawing to help show the problem.
Thanks for any insight. This isn't for school, so there's no rush or anything. Just curious.

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Oct27-13, 01:16 AM
P: 17
Oct27-13, 02:47 PM
Sci Advisor
P: 6,104
More complicated case is irrelevant. The distance between the centers (and the radii) are all that matter.

General approach: Get end points of chord which connects the points of intersection of the circles.

For each circle, calculate the angle made by connecting these end points to the center. Now the problem is reduced to a pair of problems of obtaining for each circle the area of the sector minus the area of the isosceles triangle within the sector.

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