Answer :
Let's go through the steps to find the area of the regular pentagon.
1. Determine the perimeter of the pentagon:
A regular pentagon has 5 equal sides. Given that each side has a length of [tex]\( 7 \)[/tex] cm, we can calculate the perimeter [tex]\( P \)[/tex] as:
[tex]\[ P = \text{number of sides} \times \text{side length} \][/tex]
[tex]\[ P = 5 \times 7 \][/tex]
[tex]\[ P = 35 \text{ cm} \][/tex]
2. Calculate the area of the pentagon:
The formula for the area ([tex]\( A \)[/tex]) of a regular polygon is:
[tex]\[ A = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} \][/tex]
We have already calculated the perimeter as [tex]\( 35 \)[/tex] cm, and the given apothem is [tex]\( 8 \)[/tex] cm. Plugging these values into the formula gives:
[tex]\[ A = \frac{1}{2} \times 35 \times 8 \][/tex]
[tex]\[ A = \frac{1}{2} \times 280 \][/tex]
[tex]\[ A = 140 \][/tex]
Therefore, the area of the pentagon is [tex]\( 140 \)[/tex] [tex]\( \text{cm}^2 \)[/tex].
So, the correct answer is:
[tex]\[ \boxed{140 \text{ cm}^2} \][/tex]
1. Determine the perimeter of the pentagon:
A regular pentagon has 5 equal sides. Given that each side has a length of [tex]\( 7 \)[/tex] cm, we can calculate the perimeter [tex]\( P \)[/tex] as:
[tex]\[ P = \text{number of sides} \times \text{side length} \][/tex]
[tex]\[ P = 5 \times 7 \][/tex]
[tex]\[ P = 35 \text{ cm} \][/tex]
2. Calculate the area of the pentagon:
The formula for the area ([tex]\( A \)[/tex]) of a regular polygon is:
[tex]\[ A = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem} \][/tex]
We have already calculated the perimeter as [tex]\( 35 \)[/tex] cm, and the given apothem is [tex]\( 8 \)[/tex] cm. Plugging these values into the formula gives:
[tex]\[ A = \frac{1}{2} \times 35 \times 8 \][/tex]
[tex]\[ A = \frac{1}{2} \times 280 \][/tex]
[tex]\[ A = 140 \][/tex]
Therefore, the area of the pentagon is [tex]\( 140 \)[/tex] [tex]\( \text{cm}^2 \)[/tex].
So, the correct answer is:
[tex]\[ \boxed{140 \text{ cm}^2} \][/tex]
