1. **Problem statement:** We need to find the area of a regular octagon with each side length $s = 10$ cm. 2. **Formula and approach:** A regular octagon can be divided into 8 equal isosceles triangles. The area of the octagon is 8 times the area of one such triangle. 3. **Key details:** Each triangle has two sides equal to the radius $r$ of the circumscribed circle and a base equal to the side length $s$ of the octagon. 4. **Calculate the central angle:** The central angle of each triangle is $\frac{360^\circ}{8} = 45^\circ$. 5. **Find the apothem (height) of each triangle:** The apothem $a$ is the distance from the center to the midpoint of a side. It can be found using $$a = s \times \frac{1}{2 \tan(\frac{\pi}{8})}$$ 6. **Calculate the apothem:** Using $s=10$, $$a = 10 \times \frac{1}{2 \tan(\frac{\pi}{8})} = 5 \times \frac{1}{\tan(22.5^\circ)}$$ Since $\tan(22.5^\circ) = \sqrt{2} - 1 \approx 0.4142$, then $$a = 5 \times \frac{1}{0.4142} \approx 5 \times 2.414 = 12.07 \text{ cm}$$ 7. **Calculate the area of the octagon:** The area formula for a regular polygon is $$\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}$$ The perimeter $P = 8 \times 10 = 80$ cm. So, $$\text{Area} = \frac{1}{2} \times 80 \times 12.07 = 40 \times 12.07 = 482.8 \text{ cm}^2$$ **Final answer:** The area of the regular octagon is approximately $482.8$ cm$^2$.