Please choose from the answers provided
Stephan has a box in the shape of a hexagonal prism where the hexagonal bases are regular.
He measured the height of the box to be 8 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 had a height of 9 in and a base of 10 in.
What is the surface area of the hexagonal prism?
A.
780 sq in
B.
750 sq in
C.
1,500 sq in
D.
1,020 sq in
D. 1,020 sq in
This is a triangular prism, please choose from the answers provided
Side a = 15 feet, side b = 10 feet, side c = 21 feet, side d = 18 feet, and side e = 15 feet. What is the surface area of this figure?
A.
1,053 square feet
B.
975 square feet
C.
585 square feet
D.
735 square feet
D. 735 square feet
To find the surface area of the hexagonal prism, we need to find the area of each face and then add them together.
First, let's find the area of the hexagonal bases. Since the bases are regular hexagons, we can use the formula for the area of a regular hexagon:
Area of a regular hexagon = (3√3 * side length^2) / 2
Given that the height of the prism is 8 in, we can use the height of one of the triangles formed by the line drawn from the center of the hexagon to one of its vertices to find the side length of the hexagon.
Using the formula for the area of a triangle:
Area of a triangle = (base * height) / 2
Given that the height of the triangle is 9 in and the base is 10 in, we can solve for the side length of the hexagon:
9 in = (10 in * side length) / 2
Simplifying the equation, we get:
18 in = 10 in * side length
side length = 18 in / 10 in
side length = 1.8 in
Now, we can substitute the side length into the formula for the area of a regular hexagon:
Area of a regular hexagon = (3√3 * (1.8 in)^2) / 2
Calculating the area of one of the bases, we get:
Area of one base = (3√3 * 3.24 in^2) / 2
Area of one base ≈ 5.896 in^2
Since there are two bases, we need to multiply the area of one base by 2:
Total area of the bases = 2 * 5.896 in^2
Total area of the bases ≈ 11.792 in^2
Next, let's find the area of the lateral faces.
The lateral faces are rectangles, and the base of each rectangle is the perimeter of one of the hexagons, which is 6 times the length of one side.
Perimeter of a hexagon = 6 * side length
Perimeter of a hexagon = 6 * 1.8 in
Perimeter of a hexagon = 10.8 in
The height of each rectangle is the height of the prism, which is 8 in.
Now, we can calculate the area of each lateral face:
Area of a rectangle = base * height
Area of a rectangle = 10.8 in * 8 in
Area of a rectangle = 86.4 in^2
Since there are six lateral faces, we need to multiply the area of one lateral face by 6:
Total area of the lateral faces = 6 * 86.4 in^2
Total area of the lateral faces = 518.4 in^2
Finally, to find the surface area of the hexagonal prism, we can add the area of the bases to the area of the lateral faces:
Surface area = Total area of the bases + Total area of the lateral faces
Surface area = 11.792 in^2 + 518.4 in^2
Surface area ≈ 530.192 in^2
Rounding to the nearest whole number, the surface area of the hexagonal prism is approximately 530 in^2.
So, none of the provided answer choices (A, B, C, D) are correct.