To find the area of a regular decagon (a 10-sided polygon) given its apothem and side length, follow these steps:
1. Understand Key Terms:
- Apothem: The perpendicular distance from the center to the midpoint of a side.
- Side Length: The length of one side of the decagon.
- Number of Sides (n): For a decagon, [tex]\( n = 10 \)[/tex].
2. Calculate the Perimeter:
The perimeter of a regular polygon is the length of one side multiplied by the number of sides.
[tex]\[
\text{Perimeter} = \text{side length} \times \text{number of sides}
\][/tex]
Given:
[tex]\[
\text{side length} = 5.2 \text{ meters}, \quad \text{number of sides} = 10
\][/tex]
So:
[tex]\[
\text{Perimeter} = 5.2 \times 10 = 52.0 \text{ meters}
\][/tex]
3. Calculate the Area:
The area [tex]\( A \)[/tex] 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:
[tex]\[
\text{Apothem} = 8 \text{ meters}, \quad \text{Perimeter} = 52.0 \text{ meters}
\][/tex]
So:
[tex]\[
A = \frac{1}{2} \times 52.0 \times 8 = 0.5 \times 52.0 \times 8 = 208.0 \text{ square meters}
\][/tex]
Therefore, the area of the regular decagon is:
[tex]\[
208.0 \text{ square meters}
\][/tex]
