1. **State the problem:** We need to find the radius of a circle given its circumference is $45\pi$ centimeters. 2. **Formula used:** The circumference $C$ of a circle is related to its radius $r$ by the formula: $$C = 2\pi r$$ 3. **Substitute the given circumference:** $$45\pi = 2\pi r$$ 4. **Solve for $r$:** Divide both sides by $2\pi$: $$\frac{45\pi}{2\pi} = r$$ 5. **Cancel common factors:** $$\frac{\cancel{45}\pi}{2\cancel{\pi}} = r$$ 6. **Simplify:** $$r = \frac{45}{2} = 22.5$$ **Final answer:** The radius of the circle is $22.5$ centimeters.