To find the area of a regular decagon with an apothem of 8 meters and a side length of 5.2 meters, we can follow these steps:
1. Identify the properties of the decagon:
- A decagon is a polygon with 10 equal sides and 10 equal angles.
- The apothem is the perpendicular distance from the center to a side.
2. Calculate the perimeter of the decagon:
- The perimeter [tex]\( P \)[/tex] of a polygon is the sum of the lengths of all its sides.
- Here, the length of each side is 5.2 meters, and since a decagon has 10 sides, we calculate the perimeter as:
[tex]\[
P = 10 \times 5.2 = 52 \text{ meters}
\][/tex]
3. Use the formula for the area of a regular polygon:
- The general formula for the area [tex]\( A \)[/tex] of a regular polygon is:
[tex]\[
A = \frac{1}{2} \times \text{apothem} \times \text{perimeter}
\][/tex]
- Substituting the given values:
[tex]\[
A = \frac{1}{2} \times 8 \times 52
\][/tex]
4. Perform the multiplication and division to find the area:
- First, multiply the apothem by the perimeter:
[tex]\[
8 \times 52 = 416
\][/tex]
- Then, divide by 2 to find the area:
[tex]\[
A = \frac{416}{2} = 208 \text{ square meters}
\][/tex]
Therefore, the area of the regular decagon is [tex]\( 208 \, \text{m}^2 \)[/tex].
