1. **State the problem:** We are given the circumference of a circle as $5\pi$ cm and need to find its diameter in centimeters. 2. **Formula used:** The circumference $C$ of a circle is related to its diameter $d$ by the formula: $$C = \pi d$$ 3. **Apply the formula:** Substitute $C = 5\pi$ into the formula: $$5\pi = \pi d$$ 4. **Solve for $d$:** Divide both sides by $\pi$: $$\frac{5\cancel{\pi}}{\cancel{\pi}} = \frac{\pi d}{\pi}$$ which simplifies to $$5 = d$$ 5. **Answer:** The diameter of the circle is $5$ centimeters.