To find the area of a regular decagon with an apothem of 8 meters and a side length of 5.2 meters, follow these steps:
1. Calculate the perimeter of the decagon:
A decagon has 10 sides. The perimeter (P) of a regular polygon is found by multiplying the side length (s) by the number of sides (n).
[tex]\[
P = s \times n
\][/tex]
Here, the side length [tex]\( s \)[/tex] is 5.2 meters and the number of sides [tex]\( n \)[/tex] is 10.
[tex]\[
P = 5.2 \, \text{m} \times 10 = 52 \, \text{m}
\][/tex]
2. Apply the area formula for a regular polygon:
The area (A) of a regular polygon can be calculated using the formula:
[tex]\[
A = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}
\][/tex]
Substitute the values for the perimeter (52 meters) and the apothem (8 meters) into the formula:
[tex]\[
A = \frac{1}{2} \times 52 \, \text{m} \times 8 \, \text{m}
\][/tex]
3. Calculate the area:
Perform the multiplication and division:
[tex]\[
A = \frac{1}{2} \times 416 \, \text{m}^2 = 208 \, \text{m}^2
\][/tex]
Thus, the area of the regular decagon is
[tex]\[
208 \, \text{m}^2
\][/tex]
