1. **State the problem:** We need to find the radius $r$ of a cone given its height $h=12$ cm and volume $V=36\pi$ cubic cm. 2. **Formula for the volume of a cone:** $$V = \frac{1}{3} \pi r^2 h$$ This formula relates the volume $V$, radius $r$, and height $h$ of a cone. 3. **Substitute the known values:** $$36\pi = \frac{1}{3} \pi r^2 \times 12$$ 4. **Simplify the right side:** $$36\pi = \frac{1}{3} \pi r^2 \times 12 = 4\pi r^2$$ 5. **Divide both sides by $\pi$ to cancel it out:** $$\frac{36\cancel{\pi}}{\cancel{\pi}} = 4 r^2 \implies 36 = 4 r^2$$ 6. **Divide both sides by 4:** $$\frac{36}{\cancel{4}} = \frac{4 r^2}{\cancel{4}} \implies 9 = r^2$$ 7. **Take the square root of both sides:** $$r = \sqrt{9} = 3$$ **Final answer:** The radius of the cone is $3$ centimeters.