Consider this badly drawn picture and the circle in the picture. Suppose that the circle has unit length, so that the area is $\pi$. Suppose that We know that the area enclosed by $ABCD$ is exactly $\pi/12$. Can one then find the angle $CAB$?If so, how? I tried analytic methods, but to no real help.
Let $\theta$ be the angle $CAB$ in radians. Then the area ABCD is equal to $\frac{\theta+\sin(\theta)\cos(\theta)}{2}=\frac{\pi}{12}$. Rearranging that and using $\sin(2\theta)=2\sin(\theta)\cos(\theta)$, we get that $\theta$ satisfies $2\theta+\sin(2\theta)=\frac{\pi}{3}$.
I'm not sure if there exists an analytical solution to this: Solve $\sin x = 1 - x$
However, Mathematica gives $0.2681334894944452$ as a numerical approximation.