# Simplification of ratio of intersection area of two circles to area of circles

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 $d, r, R$?

• 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.
– user856
Commented Dec 19, 2017 at 13:02
• 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. Commented Dec 19, 2017 at 17:03