To find the area of a regular polygon, such as the one described with 12 sides, an apothem of 16 meters, and a side length of 8.6 meters, we can follow these steps:
1. Determine the Perimeter of the Polygon:
The perimeter of a polygon is the sum of the lengths of all its sides. Since it is a regular polygon with 12 sides, each of which is 8.6 meters long, we can calculate the perimeter by multiplying the number of sides by the length of one side.
[tex]\[
\text{Perimeter} = \text{number of sides} \times \text{length of one side}
\][/tex]
[tex]\[
\text{Perimeter} = 12 \times 8.6 = 103.2 \text{ meters}
\][/tex]
2. Calculate the Area of the Polygon:
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]
Substituting the perimeter and apothem we calculated into the formula gives:
[tex]\[
A = \frac{1}{2} \times 103.2 \times 16
\][/tex]
Simplifying further, we get:
[tex]\[
A = 0.5 \times 103.2 \times 16 = 825.6 \text{ square meters}
\][/tex]
Therefore, the area of the regular polygon is [tex]\(825.6 \text{ m}^2\)[/tex].
