To find the area of a regular polygon, we need to follow a step-by-step approach using the given parameters. Here are the details:
1. Identify the number of sides, apothem, and side length:
- Number of sides ([tex]\(n\)[/tex]): 12
- Apothem ([tex]\(a\)[/tex]): 16 meters
- Side length ([tex]\(s\)[/tex]): 8.6 meters
2. Calculate the perimeter of the polygon:
The perimeter ([tex]\(P\)[/tex]) of a polygon is found by multiplying the number of sides by the length of each side.
[tex]\[
P = n \times s = 12 \times 8.6 = 103.2 \text{ meters}
\][/tex]
3. Use the perimeter and apothem to find the area:
The formula for the area of a regular polygon is given by:
[tex]\[
\text{Area} = \frac{1}{2} \times P \times a
\][/tex]
Substituting the calculated perimeter and given apothem:
[tex]\[
\text{Area} = \frac{1}{2} \times 103.2 \times 16
\][/tex]
[tex]\[
\text{Area} = 51.6 \times 16 = 825.6 \text{ square meters}
\][/tex]
Therefore, the area of the regular polygon with 12 sides, an apothem of 16 meters, and a side length of 8.6 meters is:
[tex]\[
825.6 \, \text{m}^2
\][/tex]
