Geometry - How do I find this angle in terms of length/radius I hope I can be clear in my request, english is not my first language.
First, take a look at this simpler diagram: only two arcs
Here is a link to my picture, need more rep apparently.

This is made of two equal circles, intersecting at two points so that the left and right sides are mirrors. IF you were to draw two lines that are tangent to the circles (at the intersection point), that would produce an angle. I found out how to get this angle in terms of the length and radius of the circles :)
Now my problem got a bit more complicated, it is the same scenario.

There is now three circles, the intersections are symmetric, the arc lengths and radius of each bolded part is the same. I haven't drawn the angles very well (it doesn't look tangent), but it should be the same as the first picture, just with an additional circle.
Again, L(1)=L(2)=L(3), R(1)=R(2)=R(3)
How do i determine the angle as a function of length and radius ?
Thanks
 A: Please consider the following diagram:
Let the angles be known by their colors, so $G=$ green angle, $P=$ purple angle, $Y=$ yellow angles, $B=$ blue angles, and $R=$ red angles. Also, let $l=L_1=L_2=L_3$ and $r=R_1=R_2=R_3$
Note that $G$ is the angle we desire.
We now make a series of observations:
First, we observe that $\frac{l}{2\pi r}=\frac{R}{2\pi}$, so $R=\frac{l}{r}$ (in radians).
Second, by symmetry, the centers form an equilateral triangle, so $B=\frac{\pi}{3}$.
Third, as the lines forming $P$ are perpendicular to the lines forming $G$, we have $G=\pi-P$
Fourth, because they form a triangle, we have $2Y+P=\pi$
And fifth, because they comprise the entire angle around a point, we have $B+2Y+R=2\pi$
Putting this all together, we can solve for $G$:
$$G=\pi-P=\pi-(\pi-2Y)=2Y=2\pi-B-R=2\pi-\frac{\pi}{3}-\frac{l}{r}=\frac{5\pi}{3}-\frac{l}{r}$$
A: If you have the solution for two circles, then for the three circle case, you just need to consider all three pairs of circles. If all three angles are the same, then you just need to apply it to any pair of adjacent circles.
In the two circle case, I assume you got $\theta = l/r$. In the 3 circle case, I believe it is $\frac{5\pi}{3}-l/r$. The reasoning is as follows. For your interior dashed arcs, they are divided into 3 sections, the outer two of which (call them $\alpha$) are equal in subtended angle, and always equal to $\pi/3$, and call the middle subtended angle $\beta$. The angle you are looking for is $\alpha+\beta$, which is equal to $2\pi-l/r-\pi/3$.
