1. **Problem Statement:** Find the area of a regular heptagon and a regular octagon given the perimeter $P=12$. 2. **Formula for area of a regular polygon:** $$\text{Area} = \frac{1}{4} n s^2 \cot\left(\frac{\pi}{n}\right)$$ where $n$ is the number of sides and $s$ is the side length. 3. **Calculate side length $s$:** Since perimeter $P=12$, and $P = n \times s$, we have $$s = \frac{P}{n} = \frac{12}{n}$$ 4. **Calculate area of the heptagon ($n=7$):** $$s_7 = \frac{12}{7}$$ $$\text{Area}_7 = \frac{1}{4} \times 7 \times \left(\frac{12}{7}\right)^2 \times \cot\left(\frac{\pi}{7}\right)$$ 5. **Calculate area of the octagon ($n=8$):** $$s_8 = \frac{12}{8} = 1.5$$ $$\text{Area}_8 = \frac{1}{4} \times 8 \times (1.5)^2 \times \cot\left(\frac{\pi}{8}\right)$$ 6. **Evaluate cotangents and simplify:** - Approximate $\cot\left(\frac{\pi}{7}\right) \approx 2.076$. - Approximate $\cot\left(\frac{\pi}{8}\right) = 1 + \sqrt{2} \approx 2.414$. 7. **Calculate numeric values:** $$\text{Area}_7 = \frac{7}{4} \times \frac{144}{49} \times 2.076 = \frac{7 \times 144 \times 2.076}{4 \times 49} \approx 10.66$$ $$\text{Area}_8 = 2 \times 2.25 \times 2.414 = 10.87$$ **Final answers:** - Area of heptagon $\approx 10.66$ - Area of octagon $\approx 10.87$