Answer :
To find the area of a regular decagon (a 10-sided polygon), we can use the formula:
[tex]\[ \text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} \][/tex]
Step-by-step solution:
1. Determine the perimeter of the decagon:
The perimeter of a regular polygon is calculated by multiplying the number of sides by the length of one side.
Given:
- Number of sides (\( n \)) = 10
- Side length (\( s \)) = 5.2 meters
So, the perimeter (\( P \)) is:
[tex]\[ P = n \times s \][/tex]
[tex]\[ P = 10 \times 5.2 \][/tex]
[tex]\[ P = 52.0 \text{ meters} \][/tex]
2. Calculate the area using the given apothem:
The apothem (\( a \)) is the perpendicular distance from the center of the polygon to the middle of one of its sides. The area (\( A \)) of a polygon can be calculated by:
[tex]\[ A = \frac{1}{2} \times P \times a \][/tex]
Given:
- Apothem (\( a \)) = 8 meters
Substitute the values of the perimeter and apothem into the formula:
[tex]\[ A = \frac{1}{2} \times 52.0 \times 8 \][/tex]
[tex]\[ A = \frac{1}{2} \times 416 \][/tex]
[tex]\[ A = 208.0 \text{ square meters} \][/tex]
Thus, the area of the regular decagon is \( 208.0 \) square meters.
[tex]\[ \boxed{208.0 \text{ m}^2} \][/tex]
[tex]\[ \text{Area} = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} \][/tex]
Step-by-step solution:
1. Determine the perimeter of the decagon:
The perimeter of a regular polygon is calculated by multiplying the number of sides by the length of one side.
Given:
- Number of sides (\( n \)) = 10
- Side length (\( s \)) = 5.2 meters
So, the perimeter (\( P \)) is:
[tex]\[ P = n \times s \][/tex]
[tex]\[ P = 10 \times 5.2 \][/tex]
[tex]\[ P = 52.0 \text{ meters} \][/tex]
2. Calculate the area using the given apothem:
The apothem (\( a \)) is the perpendicular distance from the center of the polygon to the middle of one of its sides. The area (\( A \)) of a polygon can be calculated by:
[tex]\[ A = \frac{1}{2} \times P \times a \][/tex]
Given:
- Apothem (\( a \)) = 8 meters
Substitute the values of the perimeter and apothem into the formula:
[tex]\[ A = \frac{1}{2} \times 52.0 \times 8 \][/tex]
[tex]\[ A = \frac{1}{2} \times 416 \][/tex]
[tex]\[ A = 208.0 \text{ square meters} \][/tex]
Thus, the area of the regular decagon is \( 208.0 \) square meters.
[tex]\[ \boxed{208.0 \text{ m}^2} \][/tex]
