To find the area of a regular decagon with an apothem of 5 meters and a side length of 3.25 meters, we'll follow a systematic approach using the properties of regular polygons.
### Step-by-Step Solution
1. Understand the key elements in the problem:
- Apothem (a): The distance from the center of the polygon to the midpoint of one of its sides, which is 5 meters.
- Side Length (s): The length of one side of the polygon, which is 3.25 meters.
- Number of sides (n): Since it's a decagon, it has 10 sides.
2. Calculate the perimeter (P) of the decagon:
The perimeter is the total length around the decagon. Since it has 10 sides, and each side is 3.25 meters long, the perimeter can be calculated as follows:
[tex]\[
\text{Perimeter} = \text{Number of sides} \times \text{Side length}
\][/tex]
Substituting the known values:
[tex]\[
\text{Perimeter} = 10 \times 3.25 = 32.5 \text{ meters}
\][/tex]
3. Use the formula for the area (A) of a regular polygon:
The formula to find the area of a regular polygon is given by:
[tex]\[
\text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}
\][/tex]
Using the calculated perimeter and the given apothem:
[tex]\[
\text{Area} = \frac{1}{2} \times 32.5 \times 5
\][/tex]
Simplifying this, we get:
[tex]\[
\text{Area} = \frac{1}{2} \times 162.5 = 81.25 \text{ square meters}
\][/tex]
### Final Answer
The area of the regular decagon is:
[tex]\[
81.25 \, \text{m}^2
\][/tex]
