1. **Problem statement:** We are given a sector of a circle with radius $r = 15$ cm and central angle $\theta = 120^\circ$. We want to find the perimeter of this sector. 2. **Formula for the perimeter of a sector:** The perimeter $P$ of a sector is the sum of the lengths of the two radii and the arc length. That is: $$P = 2r + s$$ where $s$ is the arc length. 3. **Formula for arc length:** The arc length $s$ of a sector with radius $r$ and central angle $\theta$ (in degrees) is: $$s = \frac{\theta}{360} \times 2\pi r$$ 4. **Calculate the arc length:** $$s = \frac{120}{360} \times 2\pi \times 15 = \frac{1}{3} \times 2\pi \times 15 = 10\pi$$ 5. **Calculate the perimeter:** $$P = 2 \times 15 + 10\pi = 30 + 10\pi$$ 6. **Approximate the perimeter:** Using $\pi \approx 3.1416$, $$P \approx 30 + 10 \times 3.1416 = 30 + 31.416 = 61.416 \text{ cm}$$ **Final answer:** The perimeter of the sector is approximately $61.42$ cm.