To find the area of a regular decagon (a 10-sided polygon) with an apothem of 5 meters and a side length of 3.25 meters, follow these steps:
1. Determine the Perimeter:
The perimeter [tex]\( P \)[/tex] of a polygon is found by multiplying the number of sides [tex]\( n \)[/tex] by the side length [tex]\( s \)[/tex]. For a decagon, [tex]\( n = 10 \)[/tex] and [tex]\( s = 3.25 \)[/tex] meters.
[tex]\[
P = n \times s = 10 \times 3.25 = 32.5 \text{ meters}
\][/tex]
2. Calculate the Area:
The area [tex]\( A \)[/tex] of a regular polygon can be calculated using the formula:
[tex]\[
A = \frac{1}{2} \times P \times a
\][/tex]
where [tex]\( a \)[/tex] is the apothem and [tex]\( P \)[/tex] is the perimeter.
Plug in the values for the perimeter and the apothem ([tex]\( a = 5 \)[/tex] meters):
[tex]\[
A = \frac{1}{2} \times 32.5 \times 5 = 0.5 \times 32.5 \times 5 = 81.25 \text{ square meters}
\][/tex]
3. Rounding the Area:
Finally, round the calculated area to the nearest tenth:
[tex]\[
81.25 \text{ rounded to the nearest tenth is } 81.2 \text{ square meters}
\][/tex]
Thus, the area of the regular decagon is [tex]\( 81.2 \)[/tex] square meters.
