A regular octagon has an apothem measuring 10 in. and a perimeter of 66.3 in.

What is the area of the octagon, rounded to the nearest square inch?

A. 88 in. [tex][tex]$^2$[/tex][/tex]
B. 175 in. [tex][tex]$^2$[/tex][/tex]
C. 332 in. [tex][tex]$^2$[/tex][/tex]
D. 700 in. [tex][tex]$^2$[/tex][/tex]



Answer :

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].