https://file.aiquickdraw.com/mathbot/file/1727060314017ohywxa5k.jpg

The problem you're working with involves calculating the area and perimeter of a regular polygon. Based on the image, the following information is given:

- Side length of the polygon: \( 8.72 \, \text{m} \)
- The polygon is a pentagon (5 sides).
- The apothem (denoted as \( L \)) is given as \( 6 \, \text{m} \).

### Perimeter Calculation:
For a regular polygon, the perimeter \( P \) is calculated as:
\[
P = n \times \text{side length}
\]
Where \( n \) is the number of sides.

Here, \( n = 5 \) and the side length is \( 8.72 \, \text{m} \), so:
\[
P = 5 \times 8.72 = 43.6 \, \text{m}
\]

### Area Calculation:
The area \( A \) of a regular polygon can be calculated using the formula:
\[
A = \frac{1}{2} \times P \times \text{apothem}
\]
Where:
- \( P \) is the perimeter
- The apothem is the distance from the center of the polygon to the middle of one of its sides.

Given:
- \( P = 43.6 \, \text{m} \)
- Apothem \( = 6 \, \text{m} \)

Thus:
\[
A = \frac{1}{2} \times 43.6 \times 6 = 130.8 \, \text{m}^2
\]

### Summary:
- **Perimeter**: \( 43.6 \, \text{m} \)
- **Area**: \( 130.8 \, \text{m}^2 \)

Let me know if you'd like any further details or clarifications.

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Here are 5 questions that expand on this topic:
1. How would the formula for area change if the polygon had more sides, such as a hexagon?
2. Can you derive the formula for the area of a regular polygon from basic geometric principles?
3. What happens to the area of a polygon as the number of sides increases?
4. How is the apothem related to the radius of the circumscribed circle in a regular polygon?
5. How would the area calculation differ for an irregular polygon?

**Tip**: Always verify the number of sides and apothem values to ensure correct calculations for polygons!

Calculate the area and perimeter of the polygon using the data from the figure attached, where each side measures 8.72 m.

Learn how to calculate the area and perimeter of a regular pentagon with a side length of 8.72 meters and an apothem of 6 meters. This guide provides formulas, a step-by-step solution, and expands your understanding with related questions.

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