Find the area of the regular polygon. Give the answer to the nearest tenth.
hexagon with a radius of 5 in.
(1 point)
Responses
65.0 in.²
65.0 in.²
129.9 in.²
129.9 in.²
259.8 in.²
259.8 in.²
53.0 in.²
To find the area of a regular polygon, you can use the formula:
Area = (1/2) x Perimeter x Apothem
In this case, since it is a hexagon, it has 6 sides, so the perimeter is 6 times the length of one side. Furthermore, the apothem is the distance from the center of the hexagon to any one of its sides. Since it is a regular hexagon, all the sides and angles are equal. The apothem can be found using the formula:
Apothem = radius x cos(30°)
Substituting the given values:
Apothem = 5 in. x cos(30°)
Apothem = 5 in. x 0.866025 (approximating cos(30°) to 6 decimal places)
Apothem ≈ 4.33 in.
Now, let's calculate the perimeter of the hexagon. Since all sides are equal, we can find one side by dividing the perimeter by 6:
Perimeter = 6 sides x side length
Perimeter = 6 x (2 x radius) (since the radius is half of the side length)
Perimeter = 6 x (2 x 5 in.)
Perimeter = 6 x 10 in.
Perimeter = 60 in.
Now, we can calculate the area using the formula:
Area = (1/2) x Perimeter x Apothem
Area = (1/2) x 60 in. x 4.33 in.
Area = 129.9 in.² (rounded to the nearest tenth)
Therefore, the correct answer is 129.9 in.².