How can I find the radius of this circle. Versine, x, and theta are known. (diagram included)

Question

diagram

So we know the versine length, and we know the length of x (blue line). We also know the angle theta. From this information it is simple to find the lengths of the dotted blue/green line (idk if we need that info or not). How can I find the radius of this circle? I am pretty sure that I can find the radius if I can find L, so maybe that is a necessary intermediate step.

Thank you in advance.

Answer 1

Lets $v$ be versine, and $R$ be the radius.

By the law of cosines.

$R^2 = (R-v)^2 + x^2 + 2(R-v)x\cos\theta$

And solve for $R$

$R = \frac {v^2 + x^2 - 2vx\cos\theta}{2v-2x\cos\theta}$

Answer 2

By Law of Cosines, $R^2 = x^2 + (R-v)^2 + 2x(R-v)\cos\theta$.

Solving this should give your solution.