To find the area of a regular octagon given its apothem and perimeter, we can use a specific formula related to polygons. The area [tex]\( A \)[/tex] of a regular polygon can be calculated using the formula:
[tex]\[ A = \frac{1}{2} \times \text{perimeter} \times \text{apothem} \][/tex]
Given:
- Apothem ([tex]\( a \)[/tex]) = 10 inches
- Perimeter ([tex]\( P \)[/tex]) = 66.3 inches
Let's calculate the area step-by-step:
1. Substitute the given values into the area formula:
[tex]\[ A = \frac{1}{2} \times 66.3 \times 10 \][/tex]
2. First, compute the product of the perimeter and the apothem:
[tex]\[ 66.3 \times 10 = 663 \][/tex]
3. Now, divide this product by 2 to find the area:
[tex]\[ \frac{663}{2} = 331.5 \][/tex]
4. To find the area rounded to the nearest square inch, we round 331.5 to the closest whole number, which is 332.
Therefore, the area of the octagon, rounded to the nearest square inch, is:
[tex]\[ \boxed{332} \text{ in}^2 \][/tex]
Thus, the correct answer is 332 in. [tex]$^2$[/tex].
