To find the area of a regular octagon, we can use the relationship between the area, the perimeter, and the apothem of the polygon.
The formula for the area [tex]\(A\)[/tex] of a regular polygon is given by:
[tex]\[
A = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}
\][/tex]
Let's go through the steps step-by-step.
1. Calculate the Perimeter:
- The perimeter [tex]\(P\)[/tex] of a regular polygon is the total length of all its sides. For a regular octagon with 8 sides, each side of length [tex]\(12.4\)[/tex] cm, the perimeter [tex]\(P\)[/tex] can be found by:
[tex]\[
P = \text{Number of sides} \times \text{Side length}
\][/tex]
[tex]\[
P = 8 \times 12.4
\][/tex]
[tex]\[
P = 99.2 \, \text{cm}
\][/tex]
2. Determine the Area:
- Using the formula for the area of a regular polygon:
[tex]\[
A = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}
\][/tex]
Given the perimeter [tex]\(P = 99.2\)[/tex] cm and the apothem [tex]\(a = 15\)[/tex] cm, we substitute these values into the formula:
[tex]\[
A = \frac{1}{2} \times 99.2 \times 15
\][/tex]
[tex]\[
A = \frac{1}{2} \times 1488
\][/tex]
[tex]\[
A = 744 \, \text{cm}^2
\][/tex]
Thus, the area of the regular octagon is:
[tex]\[
\boxed{744 \, \text{cm}^2}
\][/tex]
