1. **State the problem:** We are given the volume of a cone as $378\pi$ cm$^3$ and the radius $r=9$ cm. We need to find the missing height $h$. 2. **Formula:** The volume $V$ of a cone is given by $$V = \frac{1}{3} \pi r^2 h$$ where $r$ is the radius and $h$ is the height. 3. **Substitute known values:** $$378\pi = \frac{1}{3} \pi \times 9^2 \times h$$ 4. **Simplify the equation:** $$378\pi = \frac{1}{3} \pi \times 81 \times h$$ 5. **Divide both sides by $\pi$ to cancel it out:** $$\cancel{\pi} \times 378 = \frac{1}{3} \times 81 \times h \times \cancel{\pi}$$ which simplifies to $$378 = \frac{1}{3} \times 81 \times h$$ 6. **Multiply both sides by 3 to eliminate the fraction:** $$3 \times 378 = 81 \times h$$ $$1134 = 81h$$ 7. **Divide both sides by 81 to solve for $h$:** $$h = \frac{1134}{81}$$ 8. **Simplify the fraction:** $$h = 14$$ **Final answer:** The height $h$ of the cone is $14$ cm.