4. (10 points) As shown in the figure, the area of circle $O$ is $32, O C \perp A B, \angle A O E=\angle E O D, \angle C O F=\angle F O D$, then the area of sector $E O F$ is
4
【Analysis】We can use the relationship between the radians to first find the radian of $E O F$, which can be divided into two radians, $E O C$ and $C O F$. Using the known radian relationships, it is not difficult to find the radian of $E O F$, and the area can also be easily calculated.
【Solution】According to the analysis, from the diagram, we know that $\angle A O E = \angle E O D = \angle F O D + \angle C O F + \angle E O C$;
$$
\begin{array}{l}
\text { Also, } \angle A O E = 90^{\circ} - \angle E O C, \angle F O D = \angle C O F, \\
\therefore 90^{\circ} - \angle E O C = \angle F O D + \angle C O F + \angle E O C = 2 \times \angle C O F + \angle E O C, \\
\Rightarrow 2(\angle E O C + \angle C O F) = 90^{\circ} \Rightarrow \angle E O C + \angle C O F = 45^{\circ}
\end{array}
$$
$\Rightarrow \angle E O F = 45^{\circ}$, and since the area of circle $O$ is $32 = \pi r^{2}$,
$\therefore$ the area of $E O F$ $=\frac{45^{\circ}}{360^{\circ}} \pi r^{2} = \frac{45^{\circ}}{360^{\circ}} \times 32 = 4$.
Therefore, the answer is: 4.