1. **State the problem:** We need to find the perimeter of a sector with a central angle of $70^\circ$ and radius $8$ cm. 2. **Formula for perimeter of a sector:** The perimeter $P$ of a sector is given by the sum of the lengths of the two radii and the arc length: $$P = 2r + \text{arc length}$$ 3. **Calculate the arc length:** The arc length $L$ is a fraction of the circumference of the full circle. The circumference of a circle is $2\pi r$. The fraction is the central angle divided by $360^\circ$: $$L = \frac{\theta}{360} \times 2\pi r$$ 4. **Substitute values:** $$L = \frac{70}{360} \times 2\pi \times 8$$ 5. **Simplify the fraction:** $$L = \frac{70}{360} \times 16\pi = \frac{7}{36} \times 16\pi$$ 6. **Calculate arc length:** $$L = \frac{7 \times 16 \pi}{36} = \frac{112\pi}{36}$$ 7. **Simplify the fraction:** $$L = \frac{\cancel{112}\times \pi}{\cancel{36}} = \frac{112\pi}{36} = \frac{28\pi}{9}$$ 8. **Calculate numerical value:** Using $\pi \approx 3.1416$, $$L \approx \frac{28 \times 3.1416}{9} = \frac{87.9648}{9} \approx 9.7749 \text{ cm}$$ 9. **Calculate perimeter:** $$P = 2r + L = 2 \times 8 + 9.7749 = 16 + 9.7749 = 25.7749 \text{ cm}$$ 10. **Round to 1 decimal place:** $$P \approx 25.8 \text{ cm}$$ **Final answer:** The perimeter of the sector is approximately **25.8 cm**.