II. (20 points) As shown in Figure 4, in circle $\odot O$, chord $AB$ divides the circle into two segments with areas in the ratio $1:3$. Find the size of the central angle $\angle AOB$ (require precision to $1'$ or 0.001 radians).


Let $\odot O$ have a radius of $r$, and let $M$ be a point on the minor arc $\overparen{A B}$. The radian measure of $\overparen{A M B}$ is $\alpha$. Then
$$
\begin{array}{l}
=\frac{1}{2} a r^{2}-\frac{1}{2} r^{2} \sin \alpha \text {. } \\
\end{array}
$$

Then $\frac{1}{2} a r^{2}-\frac{1}{2} r^{2} \sin \alpha=\frac{1}{4} \pi r^{2}$, which means
$$
\alpha-\frac{\pi}{2}=\sin \alpha \text {. }
$$

Let $\beta=\alpha-\frac{\pi}{2}$, then $\beta=\cos \beta$.
Using a calculator, we find $\beta \approx 0.7391$ radians.
Thus, $\alpha=\beta+\frac{\pi}{2} \approx 2.31$ (radians).