1. The problem asks to find the area of a sector of a circle with radius $10$ cm and central angle $66^\circ$. 2. The formula for the area of a sector is: $$\text{Area} = \frac{\theta}{360^\circ} \times \pi r^2$$ where $\theta$ is the central angle in degrees and $r$ is the radius. 3. Substitute the given values: $$\text{Area} = \frac{66}{360} \times \pi \times 10^2$$ 4. Simplify the fraction: $$\frac{66}{360} = \frac{\cancel{6}11}{\cancel{6}60} = \frac{11}{60}$$ 5. So, $$\text{Area} = \frac{11}{60} \times \pi \times 100 = \frac{1100}{60} \pi$$ 6. Simplify the fraction: $$\frac{1100}{60} = \frac{\cancel{10}110}{\cancel{10}6} = \frac{110}{6} = \frac{\cancel{11}10}{\cancel{11}6} = \frac{55}{3}$$ 7. Therefore, $$\text{Area} = \frac{55}{3} \pi \approx \frac{55}{3} \times 3.1416 = 57.6$$ 8. The area of the sector is approximately $57.6$ cm$^2$ to 1 decimal place.