To determine the area of a regular decagon given an apothem of 8 meters and a side length of 5.2 meters, follow these steps.
1. Understand the properties of a decagon: A decagon is a polygon with 10 sides.
2. Calculate the perimeter of the decagon:
The perimeter (P) is the sum of the lengths of all the sides.
Given that each side length (s) is 5.2 meters and there are 10 sides (n) in a decagon:
[tex]\[
P = n \times s = 10 \times 5.2 = 52.0 \text{ meters}
\][/tex]
3. Use the formula for the area of a regular polygon:
The area (A) of a regular polygon can be found using the formula:
[tex]\[
A = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}
\][/tex]
Plug in the values we have:
[tex]\[
A = \frac{1}{2} \times 52.0 \times 8
\][/tex]
4. Perform the multiplication:
[tex]\[
A = \frac{1}{2} \times 416.0 = 208.0 \text{ square meters}
\][/tex]
Therefore, the area of the regular decagon is [tex]\(\boxed{208.0 \text{ m}^2}\)[/tex].
