I am trying to get the ratio of intersection area of two circles and area of one of the circles, but as simply as possible.

I know I can calculate the ratio as an area of intersection divided by area of single circle:

$ z_x = \frac{A'}{A_x} $

where (based on wolfram.com)

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

$A_x = {\pi}r^2$

But how to simplify this as much as possible, when I only need the ratio in the end. Is it possible to remove trigonometric functions at all? Is it perhaps somehow solely correlating to relations between $d, r, R$?

  • $\begingroup$ Dividing by $\pi r^2$ does not make much difference to the complexity of the expression, and certainly does not let you eliminate any trigonometric expressions. Sorry. $\endgroup$
    – user856
    Commented Dec 19, 2017 at 13:02
  • $\begingroup$ Ok, let's say that we will not remove any trigonometric expression. However, is it possible to get the the same result (ratio of intersection and circle area, which is on interval from 0 to 1) somehow more easily? I am trying to make a programmatic computation (Python) of this ratio as fast as possible. $\endgroup$ Commented Dec 19, 2017 at 17:03

1 Answer 1


It cannot be analytically simplified any further otherwise it would have been given already in the Wolfram reference. Only special cases can be further simplified. Also notational simplification is possible but that is about all. Even a common area computation between a circle and a cutting secant line cannot in general avoid trig. Numerical calculation is the next step.

  • $\begingroup$ At wolfram they are not discussing or describing ratios between areas, only area of intersection by itself. But I understand your answer. $\endgroup$ Commented Dec 19, 2017 at 12:58

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