The perimeter of a regular octagon is 32 inches. What is the area of the octagon? Round to the nearest whole number.

To find the area of a regular octagon with a perimeter of 32 inches, follow these steps: 1. **Determine the side length:** - A regular octagon has 8 equal sides. - Side length \( s = \frac{32 \text{ inches}}{8} = 4 \text{ inches} \). 2. **Use the area formula for a regular octagon:** \[ \text{Area} = 2(1 + \sqrt{2})s^2 \] Plugging in \( s = 4 \): \[ \text{Area} = 2(1 + \sqrt{2}) \times 4^2 = 2(1 + \sqrt{2}) \times 16 = 32(1 + \sqrt{2}) \] 3. **Calculate the numerical value:** - \( \sqrt{2} \approx 1.4142 \) - \( 1 + \sqrt{2} \approx 2.4142 \) - \( 32 \times 2.4142 \approx 77.2544 \) 4. **Round to the nearest whole number:** - \( \approx 77 \text{ square inches} \) **Final Answer:** The area of the octagon is approximately 77 square inches.