Answer :
The area of the pentagon is approximately [tex]484.3 \, \text{m}^2.[/tex]
To calculate the area of the pentagon, we first need to determine the length of the unknown side.
We can do this by applying the formula for the perimeter of a polygon, which is the sum of all its side lengths.
Thus, the perimeter of the pentagon is 25 + 95 + 60 + 55 + x, where x represents the length of the unknown side.
[tex]\[ \text{Perimeter} = 25 + 95 + 60 + 55 + x \]\\\\\text{Perimeter} = 235 + x \][/tex]
Given that the perimeter is the sum of all sides, it should equal the sum of the lengths of all five sides.
[tex]\[ \text{Perimeter} = 25 + 95 + 60 + 55 + x \][/tex]
Perimeter = 235 + x
235 + x = 25 + 95 + 60 + 55 + x
235 + x = 235 + x
Therefore, the length of the unknown side is 25 meters.
Now, to calculate the area of the pentagon, we can use the formula for the area of a regular pentagon, which is:
[tex]\[ \text{Area} = \frac{1}{4} \sqrt{5(5+2\sqrt{5})} \times s^2 \][/tex]
where \(s\) is the length of one side of the pentagon.
Substituting the given side length (s = 25) into the formula:
[tex]\[ \text{Area} = \frac{1}{4} \sqrt{5(5+2\sqrt{5})} \times 25^2 \][/tex]
[tex]\[ \text{Area} = \frac{1}{4} \sqrt{5(5+2\sqrt{5})} \times 625 \][/tex]
[tex]\[ \text{Area} = \frac{1}{4} \times 25 \times \sqrt{5(5+2\sqrt{5})} \][/tex]
[tex]\[ \text{Area} = \frac{625}{4} \times \sqrt{5(5+2\sqrt{5})} \][/tex]
[tex]\[ \text{Area} \approx 484.3 \, \text{m}^2 \][/tex]
Therefore, the area of the pentagon is approximately [tex]484.3 \, \text{m}^2.[/tex]
