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Say we have two intersecting circles we only know the areas of the circles and their intersection area. How can we find the distance between circle origins?

Looking for a generalized solution using r1,r2 and intersection area of circle A and circle B.

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hevi
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  • It's not clear to me what S(A/B) represents. Do you have a formula or method by which you can calculate the area of intersection, if you are given the radii of the two circles (and perhaps S(A/B))? – A.M. Feb 28 '22 at 22:52
  • S(A/B) is the intersection area of circle A and circle B. – hevi Mar 01 '22 at 00:10
  • What have you tried? Please show your work. – Math Lover Mar 01 '22 at 00:13
  • You can divide the intersection into two circular segments and get the intersection area from the distance between the centers. Maybe you can invert that. – Ross Millikan Mar 01 '22 at 00:13
  • @RossMillikan yes but looking for a generalized solution which only uses given r1,r2 and intersection area (or in other words area of A area of B and the intersection area). – hevi Mar 01 '22 at 00:18
  • Area of intersection of two general overlapping circles is the same question, but it also includes a formula (unfortunately incorrect) for the area in terms of the radii and distance between centers. It links to a page with correct formulas. – David K Mar 01 '22 at 02:37
  • Also see Area of overlap of two circles. – David K Mar 01 '22 at 02:39
  • @DavidK looking for a function f(r1,r2,S) which will give me the distance between circles origins – hevi Mar 01 '22 at 11:47
  • You've already defined the function. The reason for the first link is that its answer is the same as the answer I would put here: you will need to use numerical methods, which is a fancy way of saying trial and error (with some tricks to make the procedure more efficient). The key to the numerical methods is a correct formula for the area of overlap given the two radii and the distance. – David K Mar 01 '22 at 13:23

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This problem is not solvable. Not solvable in the sense that there might be many possible solutions (infinitely many, in fact).

Draw a big circle. Inside of it draw a much smaller circle. The small circle is fully contained in the larger one, so the intersection area is just the area of the smaller one. Now, move the small circle away from the large circles center, keeping it fully contained in it. The size of the circles do not change, nor does the intersection area. But the distance increases.

A different formulation of the problem might be: what is the range that the distance of the circles can take, given their sizes and intersection area?

JustANoob
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