To find the area of a regular octagon with a given apothem and side length, follow these steps:
### Step 1: Understand the Given Values
We are given:
- Apothem [tex]\( a = 7 \)[/tex] inches
- Side length [tex]\( s = 5.8 \)[/tex] inches
### Step 2: Calculate the Perimeter of the Octagon
A regular octagon has 8 sides. Therefore, the perimeter [tex]\( P \)[/tex] can be calculated as:
[tex]\[ P = 8 \times \text{side length} \][/tex]
[tex]\[ P = 8 \times 5.8 \][/tex]
[tex]\[ P = 46.4 \text{ inches} \][/tex]
### Step 3: Use the Formula for the Area of a Regular Polygon
The formula for the area [tex]\( A \)[/tex] of a regular polygon is given by:
[tex]\[ A = \frac{1}{2} \times \text{perimeter} \times \text{apothem} \][/tex]
Substitute the values we have:
[tex]\[ A = \frac{1}{2} \times 46.4 \times 7 \][/tex]
### Step 4: Perform the Multiplication
First, find [tex]\( \frac{1}{2} \times 46.4 \)[/tex]:
[tex]\[ \frac{1}{2} \times 46.4 = 23.2 \][/tex]
Then, multiply this result by the apothem:
[tex]\[ 23.2 \times 7 = 162.4 \][/tex]
### Step 5: Round the Result to the Nearest Tenth
The computed area is already at one decimal place:
[tex]\[ 162.4 \][/tex]
Therefore, the area of the regular octagon is:
[tex]\[ \boxed{162.4} \text{ in}^2 \][/tex]
This concludes that the area of a regular octagon with an apothem of 7 inches and a side length of 5.8 inches is 162.4 square inches, rounded to the nearest tenth.
