I need to find the angle $\theta_L$ in the attached sketch.

The variables I know are: $R_L$, $R_G$, and $\theta_G$. So i need a formula for $\theta_L$ in terms of these. $R_g$ is also the distance between the centres of the 2 main circles.

From trig, we can find that: $$ C = 2(R_G-d_L) \tan( \theta_G / 2 ) $$

I think for this purpose we can assume $h_G = 0$ if that makes it easier, although I think that's already been taken into account in the above trig equation for $C$.

The issue is the unknown distance between the chord and the radius of the larger $R_G$ . I'm thinking the final equation will involve a ratio between the two, but I'm unsure of how to do this.

enter image description here


If you join the two centers of these circles you can form a triangle. By using Law of sines, you get that $$\frac{R_G}{\sin (\pi-\frac{\theta_L}{2}-\frac{\theta_G}{2})}=\frac{R_L}{\sin \frac{\theta_G}{2}}$$

Maybe you can use this to get $\theta_L$ in terms of the other variables.

  • $\begingroup$ Thanks. Turns out I had to use the law of cosines as this particular triangle had a number of possible solutions given the actual values of the variables. But your answer definitely sent me in the right direction, much nicer than circular segmentation. $\endgroup$ – iagreewithjosh May 1 at 6:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.