To find the area of a regular octagon with an apothem of 15 cm and a side length of 12.4 cm, you can follow these steps:
1. Understand the properties of the octagon:
- An octagon has 8 sides.
- The apothem is the perpendicular distance from the center of the octagon to one of its sides.
2. Calculate the perimeter of the octagon:
- The perimeter [tex]\( P \)[/tex] is given by the product of the number of sides and the length of each side.
[tex]\[
P = \text{number of sides} \times \text{side length}
\][/tex]
- For our octagon:
[tex]\[
P = 8 \times 12.4 \, \text{cm}
\][/tex]
- This simplifies to:
[tex]\[
P = 99.2 \, \text{cm}
\][/tex]
3. Use the formula for the area of a regular polygon:
- The formula to calculate the area [tex]\( A \)[/tex] of a regular polygon is:
[tex]\[
A = \frac{1}{2} \times P \times a
\][/tex]
- Where [tex]\( P \)[/tex] is the perimeter and [tex]\( a \)[/tex] is the apothem.
4. Substitute the known values into the formula:
- Here, [tex]\( P = 99.2 \, \text{cm} \)[/tex] and [tex]\( a = 15 \, \text{cm} \)[/tex].
[tex]\[
A = \frac{1}{2} \times 99.2 \, \text{cm} \times 15 \, \text{cm}
\][/tex]
- This simplifies to:
[tex]\[
A = \frac{1}{2} \times 1488 \, \text{cm}^2
\][/tex]
- Which further simplifies to:
[tex]\[
A = 744 \, \text{cm}^2
\][/tex]
Hence, the area of the regular octagon is [tex]\( 744 \, \text{cm}^2 \)[/tex].
