1. **State the problem:** Find the volume of a cylinder with radius $r=10$ cm and height $h=20$ cm. 2. **Formula:** The volume $V$ of a cylinder is given by the formula: $$V = \pi r^2 h$$ where $r$ is the radius and $h$ is the height. 3. **Substitute values:** $$V = \pi \times 10^2 \times 20$$ 4. **Calculate:** $$V = \pi \times 100 \times 20 = 2000\pi$$ 5. **Approximate using $\pi \approx 3.1416$:** $$V \approx 2000 \times 3.1416 = 6283.2$$ 6. **Round to nearest hundredth:** $$V \approx 6283.19$$ cubic centimeters. **Final answer:** The volume of the cylinder is approximately $6283.19$ cubic centimeters.