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Find the area of a regular polygon with 9 sides, an apothem of 38.5 cm, and a side length of 28 cm.

[tex]? \, \text{cm}^2[/tex]

Answer :

To find the area of a regular polygon with 9 sides, an apothem of 38.5 cm, and a side length of 28 cm, we'll follow these steps:

1. Calculate the Perimeter:
- The perimeter of a polygon is the total length around the polygon. For a regular polygon (which has equal-length sides), you can calculate the perimeter by multiplying the number of sides (9 in this case) by the length of each side (28 cm).
- Perimeter = Number of sides × Side length = 9 × 28 = 252 cm

2. Use the Area Formula for a Regular Polygon:
- The area [tex]\(A\)[/tex] of a regular polygon can be found using the formula:
[tex]\[
A = \frac{\text{Perimeter} \times \text{Apothem}}{2}
\][/tex]
- Here, the apothem is the perpendicular distance from the center of the polygon to the midpoint of one of its sides, which is given as 38.5 cm.

3. Calculate the Area:
- Substitute the known values into the area formula:
[tex]\[
A = \frac{252 \times 38.5}{2}
\][/tex]
- Simplifying the calculation gives:
[tex]\[
A = \frac{9696}{2} = 4851 \, \text{cm}^2
\][/tex]

Therefore, the area of this regular polygon is [tex]\(4851 \, \text{cm}^2\)[/tex].

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