How do I find the angle of intersecting circles?

Question

I'm a software engineer not a mathematician so I apologize if I'm not using proper language or if this was answered before (I couldn't find it).

I have two circles. I know the positions and radius of both. Also, circle 1 is always bigger than circle 2 and circle 2's center is always on circle 1's edge. Because of this there will always be two intersecting points. How do I find the angle of those intersecting points from circle 1?

Thank you.

Answer 1

Lets call your target angle $\theta$ The radius of the big circle $r_1$ and the small circle $r_2$

$2 r_1 \sin \frac \theta4 = r_2$

How did I get that? construct radii from the big circle to the center of the small circle and to one of the intersection points.

the angle is $\frac 12 \theta$

Now bisect that.

you have a right triangle with hypotenuse $r_1$ and the opposite leg equals $r_1\sin \frac {\theta}4$

It is also $\frac 12 r_2$