Question
Calculate a right hexagonal prism with a height of 4 units and a base of 30 units
Answers
GPT 3.5
To calculate the surface area of a right hexagonal prism, we need to find the area of the six faces and add them together.
The base of the prism is a regular hexagon, which can be split into six equilateral triangles. The formula for the area of an equilateral triangle is:
Area = (sqrt(3) / 4) * side^2
In this case, the side length of each equilateral triangle is equal to the side length of the hexagon base, which is 30 units. So the area of each equilateral triangle is:
Area = (sqrt(3) / 4) * 30^2
Area = 450 sqrt(3)
Since there are six equilateral triangles making up the base, the total area of the base is:
Total base area = 6 * 450 sqrt(3)
Total base area = 2700 sqrt(3)
The lateral surface area of the prism is the perimeter of the base multiplied by the height. The formula for the perimeter of a regular hexagon with sides of length s is:
Perimeter = 6s
So in this case, the perimeter of the base is:
Perimeter = 6 * 30
Perimeter = 180 units
The lateral surface area is:
Lateral surface area = Perimeter * height
Lateral surface area = 180 * 4
Lateral surface area = 720 units^2
Finally, to find the total surface area of the prism, we add the base area and lateral surface area:
Total surface area = Total base area + Lateral surface area
Total surface area = 2700 sqrt(3) + 720
Total surface area ≈ 3206.26 units^2
Therefore, the total surface area of the right hexagonal prism is approximately 3206.26 square units.
The base of the prism is a regular hexagon, which can be split into six equilateral triangles. The formula for the area of an equilateral triangle is:
Area = (sqrt(3) / 4) * side^2
In this case, the side length of each equilateral triangle is equal to the side length of the hexagon base, which is 30 units. So the area of each equilateral triangle is:
Area = (sqrt(3) / 4) * 30^2
Area = 450 sqrt(3)
Since there are six equilateral triangles making up the base, the total area of the base is:
Total base area = 6 * 450 sqrt(3)
Total base area = 2700 sqrt(3)
The lateral surface area of the prism is the perimeter of the base multiplied by the height. The formula for the perimeter of a regular hexagon with sides of length s is:
Perimeter = 6s
So in this case, the perimeter of the base is:
Perimeter = 6 * 30
Perimeter = 180 units
The lateral surface area is:
Lateral surface area = Perimeter * height
Lateral surface area = 180 * 4
Lateral surface area = 720 units^2
Finally, to find the total surface area of the prism, we add the base area and lateral surface area:
Total surface area = Total base area + Lateral surface area
Total surface area = 2700 sqrt(3) + 720
Total surface area ≈ 3206.26 units^2
Therefore, the total surface area of the right hexagonal prism is approximately 3206.26 square units.