1. **State the problem:** Find the total surface area of a cylinder with height $20$ cm and radius $8$ cm. 2. **Formula:** The total surface area $A$ of a cylinder is given by: $$A = 2\pi r^2 + 2\pi r h$$ where $r$ is the radius and $h$ is the height. 3. **Explanation:** The formula consists of two parts: the area of the two circular bases $2\pi r^2$ and the area of the curved surface $2\pi r h$. 4. **Substitute values:** $$A = 2\pi (8)^2 + 2\pi (8)(20)$$ 5. **Calculate:** $$A = 2\pi (64) + 2\pi (160)$$ $$A = 128\pi + 320\pi$$ 6. **Combine terms:** $$A = (128 + 320)\pi = 448\pi$$ 7. **Approximate:** Using $\pi \approx 3.1416$, $$A \approx 448 \times 3.1416 = 1406.48$$ **Final answer:** The total surface area of the cylinder is approximately $1406.48$ cm$^2$.