1. **State the problem:** Find the radius $r$ of a cylinder given its lateral area $18\pi$ and height $h=6$. 2. **Formula for lateral area of a cylinder:** $$\text{Lateral Area} = 2\pi r h$$ 3. **Substitute known values:** $$18\pi = 2\pi r \times 6$$ 4. **Simplify the equation:** $$18\pi = 12\pi r$$ 5. **Divide both sides by $12\pi$ to isolate $r$:** $$r = \frac{18\pi}{12\pi}$$ $$r = \frac{\cancel{18}\pi}{\cancel{12}\pi} = \frac{3}{2}$$ 6. **Final answer:** $$r = \frac{3}{2} = 1.5$$