1. **State the problem:** We are given the volume of a cylinder as 565.2 cm$^3$ and the height as 20 cm. We need to find the diameter of the cylinder. 2. **Formula for the volume of a cylinder:** $$V = \pi r^2 h$$ where $V$ is volume, $r$ is radius, and $h$ is height. 3. **Substitute the known values:** $$565.2 = \pi r^2 \times 20$$ 4. **Isolate $r^2$:** $$r^2 = \frac{565.2}{\pi \times 20}$$ 5. **Simplify the denominator:** $$r^2 = \frac{565.2}{20\pi}$$ 6. **Calculate $r^2$ numerically:** $$r^2 = \frac{565.2}{62.8319} \approx 9$$ 7. **Find the radius $r$ by taking the square root:** $$r = \sqrt{9} = 3 \text{ cm}$$ 8. **Find the diameter $d$:** $$d = 2r = 2 \times 3 = 6 \text{ cm}$$ **Final answer:** The diameter of the cylinder is 6 cm.