1. **State the problem:** We want to find the area of a regular octagon, which is a polygon with 8 equal sides and 8 equal angles. 2. **Formula used:** The area $A$ of a regular polygon with $n$ sides of length $s$ can be found using the formula: $$A = \frac{1}{4} n s^2 \cot\left(\frac{\pi}{n}\right)$$ For an octagon, $n=8$. 3. **Apply the formula for an octagon:** $$A = \frac{1}{4} \times 8 \times s^2 \times \cot\left(\frac{\pi}{8}\right)$$ 4. **Simplify the expression:** $$A = 2 s^2 \cot\left(\frac{\pi}{8}\right)$$ 5. **Explanation:** - The cotangent function relates to the angle inside the polygon. - This formula works because the octagon can be divided into 8 isosceles triangles, each with vertex angle $\frac{2\pi}{8} = \frac{\pi}{4}$. 6. **Alternative formula using apothem $a$:** If the apothem (distance from center to midpoint of a side) $a$ is known, the area is: $$A = \frac{1}{2} \times \text{Perimeter} \times a = \frac{1}{2} \times 8s \times a = 4 s a$$ 7. **Summary:** To find the area of a regular octagon, measure the side length $s$ and use: $$A = 2 s^2 \cot\left(\frac{\pi}{8}\right)$$ This gives the exact area in terms of $s$.