Answer :
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]
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]