1. **State the problem:** Calculate the area of a sector of a circle with radius $20$ cm and central angle $300^\circ$. 2. **Formula:** The area $A$ of a sector with radius $r$ and central angle $\theta$ (in degrees) is given by: $$ A = \frac{\theta}{360} \times \pi r^2 $$ 3. **Substitute values:** Here, $r = 20$ cm and $\theta = 300^\circ$. $$ A = \frac{300}{360} \times \pi \times 20^2 $$ 4. **Simplify the fraction:** $$ A = \frac{\cancel{300}}{\cancel{360}} \times \pi \times 400 = \frac{5}{6} \times \pi \times 400 $$ 5. **Calculate the area:** $$ A = \frac{5}{6} \times 400 \pi = \frac{2000}{6} \pi = \frac{1000}{3} \pi $$ 6. **Final answer:** $$ A = \frac{1000}{3} \pi \approx 1047.2 \text{ cm}^2 $$ So, the area of the sector is approximately $1047.2$ square centimeters.