An illustration shows a table in the shape of a 3-dimensional hexagonal prism, and its 2-dimensional unfolded version. The 3-dimensional prism shows a hexagon tabletop and table bottom with six vertical rectangular sides connecting the six edges of each hexagon to the other, and also connected to each other. The unfolded version shows 6 vertical rectangles connected to one another horizontally with a vertical length of 75 centimeters. Two identical hexagons are adjoined to the top and bottom of the first vertical rectangle. The diagonal of the top hexagon, shown as a dotted line, is labeled as 90 centimeters. A side of the bottom hexagon is labeled as 45 centimeters, and the perpendicular height from the center of the hexagon to the middle of a side is labeled as 38.97 centimeters. The perpendicular height is denoted by a right angle symbol. First Choice Interior makes a coffee table in the shape of a regular hexagonal prism. The top is made of wood, the sides of glass, and the bottom of metal. Use the net to find how many cm2 of glass are needed for the furniture company to make one coffee table.
answer choices:
30,772 cm2
20,250 cm2
17,550 cm2
40,500 cm2
The total surface area of the 3-dimensional hexagonal prism can be calculated by adding the areas of the top hexagon, the bottom hexagon, and the six rectangular sides.
The formula for the area of a regular hexagon is (3√3 × s^2)/2, where s is the length of a side.
For the top hexagon, we have s = 90 cm.
Area of the top hexagon = (3√3 × 90^2)/2 = 23,382 cm^2
For the bottom hexagon, we have s = 45 cm.
Area of the bottom hexagon = (3√3 × 45^2)/2 = 5,845.5 cm^2
For each of the six rectangular sides, we have a width of 45 cm and a height of 38.97 cm.
Area of each rectangular side = 45 × 38.97 = 1,753.65 cm^2
Total area of all six rectangular sides = 1,753.65 × 6 = 10,521.90 cm^2
Therefore, the total surface area of the hexagonal prism is:
23,382 + 5,845.5 + 10,521.90 = 39,749.40 cm^2
Since the sides of the coffee table are made of glass, the total area of glass needed for one coffee table is 10,521.90 cm^2.
So, the answer is 10,521.90 cm^2.
An illustration shows a net diagram. The center of the diagram is a triangle. Each side of the center triangle forms the base of one of three triangles extending outward from the center triangle. The center triangular base has a base edge measuring 12 centimeters, two side edges measuring 10 centimeters, and a perpendicular height measuring 8 centimeters. Its sides are marked by dashed lines, which form the bases of three connected triangles. One of the connected triangles has a base measuring 12 centimeters and a perpendicular height of 7.8 centimeters. The other two connected triangles have bases measuring 10 centimeters and perpendicular heights measuring 8 centimeters.
Which of the following is the lateral surface area of the triangular prism represented with a net?
174.8 cm2
126.8 cm2
120 cm2
253.6 cm2
To find the lateral surface area of the triangular prism represented by the net, we need to find the areas of the three outer triangles and add them together.
For the first outer triangle with a base of 12 cm and a perpendicular height of 7.8 cm:
Area = 1/2 * base * height = 1/2 * 12 * 7.8 = 46.8 cm^2
For the second outer triangle with a base of 10 cm and a perpendicular height of 8 cm:
Area = 1/2 * base * height = 1/2 * 10 * 8 = 40 cm^2
For the third outer triangle with a base of 10 cm and a perpendicular height of 8 cm:
Area = 1/2 * base * height = 1/2 * 10 * 8 = 40 cm^2
Total lateral surface area = Area of first outer triangle + Area of second outer triangle + Area of third outer triangle
Total lateral surface area = 46.8 + 40 + 40 = 126.8 cm^2
Therefore, the lateral surface area of the triangular prism represented by the net is 126.8 cm^2.
So, the answer is 126.8 cm^2.
An illustration shows a net diagram. The center of the diagram is a square with dashed lines marking its sides. Each side forms the base of one of four triangles extending outward from the square. The sides of the square measure 3 inches, and the perpendicular height of the triangular faces measures 5 inches.
A chocolate dessert is in the shape of a square pyramid. The pyramid is represented with a net. How many square inches of icing are needed for the faces of the dessert?
69 in.2
30 in.2
60 in.2
39 in.2
To find the total surface area of the square pyramid, we need to add the areas of the four triangular faces together.
Each triangular face of the square pyramid corresponds to one of the triangles in the net diagram.
For a triangular face with a base of 3 inches and a perpendicular height of 5 inches:
Area = 1/2 * base * height = 1/2 * 3 * 5 = 7.5 in^2
Since there are four triangular faces:
Total surface area = 4 * 7.5 = 30 in^2
Therefore, 30 square inches of icing are needed for the faces of the dessert.
So, the answer is 30 in^2.
The roof of a gazebo is in the shape of an octagonal pyramid. Each side of the octagon is 12 feet, the height of each triangular face is 10 feet, and the area of the octagon is 695 square feet. How many square feet of cedar are needed to cover the lateral surface area of the octagonal pyramid?(1 point)
To find the lateral surface area of the octagonal pyramid, we need to calculate the area of each of the eight triangular faces and then add them together.
The lateral surface area of a pyramid is given by the formula:
Lateral Surface Area = (1/2) * Perimeter of the base * Slant height
The perimeter of the base of the octagon is 8 * 12 = 96 feet (since each side is 12 feet).
Given that the slant height of each triangular face is 10 feet, we can calculate the lateral surface area of one triangular face:
Lateral Surface Area of one triangular face = (1/2) * 96 * 10 = 480 square feet
Since there are 8 triangular faces:
Total lateral surface area = 8 * 480 = 3840 square feet
Therefore, 3840 square feet of cedar are needed to cover the lateral surface area of the octagonal pyramid.
The roof of a gazebo is in the shape of an octagonal pyramid. Each side of the octagon is 12 feet, the height of each triangular face is 10 feet, and the area of the octagon is 695 square feet. How many square feet of cedar are needed to cover the lateral surface area of the octagonal pyramid?(1 point)
960 ft.2
1,175 ft.2
480 ft.2
360 ft.2
To find the lateral surface area of the octagonal pyramid, we need to calculate the area of each of the eight triangular faces and then add them together.
First, we need to find the perimeter of the octagonal base. Since each side of the octagon is 12 feet, the perimeter of the octagon is 8 * 12 = 96 feet.
The area of the octagon is given as 695 square feet.
The lateral surface area of a pyramid is given by the formula:
Lateral Surface Area = (1/2) * Perimeter of the base * Slant height
We need to find the slant height of the triangular faces. Using the Pythagorean theorem, the slant height can be calculated as √(10^2 + (12/2)^2) = √(100 + 36) = √136 = 11.66 feet.
Now, we can calculate the lateral surface area of one triangular face:
Lateral Surface Area of one triangular face = (1/2) * 96 * 11.66 = 559.68 square feet
Since there are 8 triangular faces:
Total lateral surface area = 8 * 559.68 = 4477.44 square feet
Therefore, 4477.44 square feet of cedar are needed to cover the lateral surface area of the octagonal pyramid.
The closest option is 480 ft^2, so the answer will be 480 ft^2.