Stephan has a box in the shape of a hexagonal prism where the hexagonal bases are regular. A net of the box is below.
Note: Figure is not drawn to scale.
He measured the height of the box to be 4 in. Then, Stephan drew a line from the center of one of the hexagons to each of its vertices and noticed that all the triangles he created have a height of 5 in and a base of 6 in.
Note: Figure is not drawn to scale.
What is the surface area of the hexagonal prism?
To find the surface area of the hexagonal prism, we need to find the area of the two hexagonal bases and the area of the six rectangular faces.
First, let's find the area of one of the hexagonal bases. Since the hexagon is regular, we can use the formula for the area of a regular hexagon:
Area of one hexagonal base = (3√3/2) * (side length)^2
Area of one hexagonal base = (3√3/2) * 6^2
Area of one hexagonal base = 3√3 * 36
Area of one hexagonal base = 108√3 square inches
Since there are two hexagonal bases, the total area of the two hexagonal bases is:
Total area of the two hexagonal bases = 2 * 108√3
Total area of the two hexagonal bases = 216√3 square inches
Next, let's find the area of one of the rectangular faces. Since the height of the triangles formed by drawing lines from the center of the hexagon to its vertices is 5 in and the base is 6 in, the height of the rectangular face is 5 in and the width is 6 in. The area of one rectangular face is:
Area of one rectangular face = height * width
Area of one rectangular face = 5 * 6
Area of one rectangular face = 30 square inches
Since there are six rectangular faces, the total area of the six rectangular faces is:
Total area of the six rectangular faces = 6 * 30
Total area of the six rectangular faces = 180 square inches
Finally, to find the total surface area of the hexagonal prism, we add the total area of the two hexagonal bases and the total area of the six rectangular faces:
Total surface area = Total area of the two hexagonal bases + Total area of the six rectangular faces
Total surface area = 216√3 + 180
Total surface area ≈ 370.8 + 180
Total surface area ≈ 550.8 square inches
Therefore, the surface area of the hexagonal prism is approximately 550.8 square inches.