To find the area of a regular octagon with an apothem of 7 inches and a side length of 5.8 inches, follow these steps:
1. Understand the given information:
- Apothem ([tex]\(a\)[/tex]): 7 inches
- Side length ([tex]\(s\)[/tex]): 5.8 inches
- Number of sides ([tex]\(n\)[/tex]): 8 (since it is a regular octagon)
2. Calculate the perimeter ([tex]\(P\)[/tex]) of the octagon:
The perimeter of a regular polygon is given by the formula:
[tex]\[
P = n \times s
\][/tex]
Substituting the values, we get:
[tex]\[
P = 8 \times 5.8
\][/tex]
Calculating this,
[tex]\[
P = 46.4 \, \text{inches}
\][/tex]
3. Calculate the area ([tex]\(A\)[/tex]) of the octagon:
The area of a regular polygon can be calculated using the formula:
[tex]\[
A = \frac{1}{2} \times P \times a
\][/tex]
Substituting the values, we have:
[tex]\[
A = \frac{1}{2} \times 46.4 \times 7
\][/tex]
Calculating this,
[tex]\[
A = \frac{1}{2} \times 324.8
\][/tex]
[tex]\[
A = 162.4 \, \text{square inches}
\][/tex]
So, the area of the regular octagon is [tex]\(162.4 \, \text{in}^2\)[/tex].
