1. Given a circle with circumference $44$ cm, find the radius $r$ using the formula for circumference: $$C=2\pi r$$ 2. Substitute $C=44$: $$44=2\pi r$$ 3. Solve for $r$: $$r=\frac{44}{2\pi}=\frac{22}{\pi}$$ 4. The problem states a sector with a central angle of $90^\circ$ is cut out. The length of the arc (the "விற்கின் நீளம்") of this sector is given by: $$L=\frac{\theta}{360^\circ} \times 2\pi r$$ where $\theta=90^\circ$. 5. Substitute values: $$L=\frac{90}{360} \times 2\pi \times \frac{22}{\pi}$$ 6. Simplify: $$L=\frac{1}{4} \times 2 \times 22=\frac{1}{4} \times 44=11$$ 7. Therefore, the length of the arc of the cut-out sector is $11$. Final answer: $11$