We wish to consider the union of all rectangles $A B C D$ with $A B=4, B C=5$, and $B C \| O M$, with center $X$ on $\Omega$. Consider translating rectangle $A B C D$ along the radius $X O$ to a rectangle $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ now centered at $O$. It is now clear that that every point inside $A B C D$ is a translate of a point in $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$, and furthermore, any rectangle $A B C D$ translates along the appropriate radius to the same rectangle $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$. We see that the boundary of this region can be constructed by constructing a quarter-circle at each vertex, then connecting these quarter-circles with tangents to form four rectangular regions. Now, splitting our region in to four quarter circles and five rectangles, we compute the desired area to be $4 \cdot \frac{1}{4}(8)^{2} \pi+2(4 \cdot 8)+2(5 \cdot 8)+(4 \cdot 5)=164+64 \pi$