To find the area of a regular polygon with 12 sides, an apothem of 16 meters, and a side length of 8.6 meters, follow these steps:
### Step 1: Calculate the Perimeter
First, calculate the perimeter of the polygon. The perimeter (P) is found by multiplying the number of sides (n) by the length of each side (s):
[tex]\[ P = n \times s \][/tex]
In this case:
[tex]\[ n = 12 \][/tex]
[tex]\[ s = 8.6 \, \text{meters} \][/tex]
So,
[tex]\[ P = 12 \times 8.6 = 103.2 \, \text{meters} \][/tex]
### Step 2: Use the Area Formula for a Regular Polygon
The formula for the area (A) of a regular polygon is:
[tex]\[ A = 0.5 \times P \times a \][/tex]
where [tex]\( P \)[/tex] is the perimeter and [tex]\( a \)[/tex] is the apothem.
Here,
[tex]\[ a = 16 \, \text{meters} \][/tex]
Substitute the known values into the formula:
[tex]\[ A = 0.5 \times 103.2 \times 16 \][/tex]
### Step 3: Perform the Multiplication
First, calculate the product of the perimeter and the apothem:
[tex]\[ 103.2 \times 16 = 1651.2 \][/tex]
Then, multiply by 0.5:
[tex]\[ 0.5 \times 1651.2 = 825.6 \, \text{square meters} \][/tex]
### Step 4: Round the Area to the Nearest Tenth
Finally, the calculated area is already to the nearest tenth:
[tex]\[ 825.6 \, \text{square meters} \][/tex]
### Conclusion
The area of the regular polygon is:
[tex]\[ 825.6 \, \text{square meters} \][/tex]
