A regular pentagon's sides each have a length of [tex][tex]$7 cm$[/tex][/tex] and an apothem of [tex][tex]$8 cm$[/tex][/tex]. What is the area of the pentagon?

A. [tex][tex]$127 cm^2$[/tex][/tex]
B. [tex][tex]$146 cm^2$[/tex][/tex]
C. [tex][tex]$140 cm^2$[/tex][/tex]
D. [tex][tex]$150 cm^2$[/tex][/tex]



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]