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I have two circles with radii $r$ and $R$. I need to know how far apart to draw them so that their intersection has area $A$. Basically, I'm solving for $d$ in the diagram seen in the MathWorld link.

It's not hard to solve for $A$ (see MathWorld),

$A = r^2\cos^{-1}\left(\frac{d^2+r^2-R^2}{2dr}\right) + R^2\cos^{-1}\left(\frac{d^2+R^2-r^2}{2dR}\right)-\frac{1}{2}\sqrt{(-d+r+R)(d+r-R)(d-r+R)(d+r+R)}$

but I've been struggling to find a solution for $d$.

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You will not find a "formula" for $d$ in terms of $R$ and $r$; for sure I can't. –  André Nicolas May 3 '12 at 15:56
I agree with André, there might not be an explicit formula. I tried calculating $A$ by some other method which gives a simpler result but at the price of two new intertwined parameters (two angles). Although different, this reminds me a bit of shapes called "lunules", which are like crescent moons. –  Olivier Bégassat May 3 '12 at 17:00

1 Answer 1

Equations arising from the areas of circular segments are often not solvable with elementary functions, largely because the area of a circular segment with radius $r$ and central angle $\theta$ is $\frac{1}{2}r^2(\theta-\sin\theta)$, which involves both $\theta$ and $\sin\theta$ in a way that can't be pulled apart algebraically. It's quite likely, though, that given specific values of your parameters $r$, $R$, and $A$ it would be possible to find a numeric approximation for $d$.

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