An illustration shows two views of a triangular prism.
The first view shows a 3-dimensional triangular prism with the highlighted rectangular base labeled as 14 inches in length and 12 inches in width. A right triangular side is labeled 5 inches in perpendicular height and is denoted by a right angle symbol at each end where the base and side meet. A hypotenuse is labeled as 13 inches. Edges that are not visible are drawn in dashed lines. The unfolded version shows three adjacent vertical rectangles where the first and the last are similar and the middle rectangle is bigger. The left rectangle is labeled 14 inches in vertical length. Two right triangles are adjoined on the top and bottom of the middle rectangle. The triangle base, which is also the width of the middle rectangle, is labeled as 12 inches. The hypotenuse of the triangle is 13 inches. The height of the right angle side of the triangle is labeled as 5.
Find the lateral surface area of the triangular prism. Solve this problem by determining the area of the shape’s net.
(1 point)
Responses
480 in.2
480 in. squared
420 in.2
420 in. squared
564 in.2
564 in. squared
504 in.2
504 in. squared
To find the lateral surface area of the triangular prism, we need to determine the total surface area of the shape's net and then subtract the area of the two triangular bases.
First, let's find the area of the net:
- The rectangular base has dimensions 14 inches by 12 inches, so the area of the base is 14 * 12 = 168 square inches.
- The two right triangles on the side each have an area of 0.5 * base * height = 0.5 * 12 * 5 = 30 square inches each.
- The middle rectangle has dimensions 14 inches by 13 inches (which is the hypotenuse of the right triangle), so the area of the middle rectangle is 14 * 13 = 182 square inches.
Adding the areas of the three shapes together: 2(30) + 168 + 182 = 420 square inches.
Finally, the lateral surface area of the triangular prism is the total surface area of the net minus the areas of the two triangular bases:
420 - (2 * 30) = 420 - 60 = 360 square inches.
Therefore, the lateral surface area of the triangular prism is 360 in.².
So the answer is:
420 in.²
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.
(1 point)
Responses
40,500 cm2
40,500 cm squared
30,772 cm2
30,772 cm squared
20,250 cm2
20,250 cm squared
17,550 cm2
To find how many square centimeters of glass are needed for First Choice Interior to make one coffee table in the shape of a regular hexagonal prism, we need to calculate the lateral surface area of the prism, which represents the glass sides of the table.
From the information provided in the unfolded net:
- The vertical length of each rectangle (which represents the lateral side of the prism) is 75 cm.
- The total lateral surface area of the hexagonal prism would be the sum of the six rectangles' areas.
Since the prism has a regular hexagonal shape, the side length of the hexagon is the same as the side length of the hexagons on the top and bottom (45 cm).
To calculate the lateral surface area, we use the formula for the lateral surface area of a hexagonal prism:
Lateral surface area = 6 * side length * height
For this case, the side length is 45 cm. The height of each lateral side is given as 75 cm.
Lateral surface area = 6 * 45 cm * 75 cm
Lateral surface area = 20250 cm²
Therefore, the furniture company needs 20,250 cm² of glass to make one coffee table.
The correct answer is:
20,250 cm²
Lateral Surface Area Quick Check
3 of 5 3 of 5 Items
Question
Use the image to answer the question.
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?
(1 point)
Responses
126.8 cm2
126.8 cm squared
174.8 cm2
174.8 cm squared
120 cm2
120 cm squared
253.6 cm2
253.6 cm squared
To find the lateral surface area of the triangular prism represented by the net, we need to calculate the total surface area of the net and then subtract the areas of the triangular bases.
First, let's find the surface area of each triangular face with the given dimensions:
1. For the center triangle:
- Base = 12 cm
- Perpendicular Height = 8 cm
Surface Area = 0.5 * base * height = 0.5 * 12 * 8 = 48 cm²
2. For the two triangles with base 10 cm and height 8 cm:
Surface Area = 0.5 * 10 * 8 = 40 cm²
3. For the triangle with base 12 cm and height 7.8 cm:
Surface Area = 0.5 * 12 * 7.8 = 46.8 cm²
Now, the total lateral surface area of the prism is the sum of the areas of the three outer triangles:
Total Lateral Surface Area = 40 + 40 + 46.8 = 126.8 cm²
Therefore, the lateral surface area of the triangular prism represented by the net is 126.8 cm².
The correct answer is:
126.8 cm²
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?
(1 point)
Responses
69 in.2
69 in. squared
39 in.2
39 in. squared
60 in.2
60 in. squared
30 in.2
To find the total surface area of the square pyramid represented by the net, we need to calculate the sum of the areas of the four triangular faces.
The net consists of four triangles, each with a base of the square (3 inches) and a perpendicular height of 5 inches.
First, let's calculate the area of one of the triangular faces:
Area of one triangular face = 0.5 * base * height
Area of one triangular face = 0.5 * 3 * 5
Area of one triangular face = 7.5 square inches
Since there are four identical triangular faces:
Total surface area of the square pyramid = 4 * Area of one triangular face
Total surface area of the square pyramid = 4 * 7.5
Total surface area of the square pyramid = 30 square inches
Therefore, 30 square inches of icing are needed for the faces of the dessert.
The correct answer is:
30 in.²
Lateral Surface Area Quick Check
5 of 5 5 of 5 Items
Question
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)
Responses
1,175 ft.2
1,175 ft. squared
360 ft.2
360 ft. squared
480 ft.2
480 ft. squared
960 ft.2
960 ft. squared
To find the lateral surface area of the octagonal pyramid, we first need to calculate the lateral surface area of one of the triangular faces and then multiply it by the number of triangular faces to cover the entire lateral surface.
The lateral surface area of a triangular face of the pyramid can be calculated using the formula:
Lateral surface area of a triangular face = 0.5 * perimeter of the base * slant height
Since the base of the octagon has sides that are 12 feet each, the perimeter of the base is 8 * 12 = 96 feet.
To find the slant height, we can use the Pythagorean theorem which states that the slant height, height, and half the side of the base form a right triangle. So, the slant height can be found as:
slant height = √(height² + (side length/2)²)
slant height = √(10² + (12/2)²)
slant height = √(100 + 36)
slant height = √136 ≈ 11.66 feet
Now, we can calculate the lateral surface area of one triangular face:
Lateral surface area of one triangular face = 0.5 * 96 * 11.66
Lateral surface area of one triangular face ≈ 559.68 square feet
Since there are 8 triangular faces on the pyramid, we multiply the lateral surface area of one face by 8:
Total lateral surface area = 8 * 559.68
Total lateral surface area ≈ 4,477.44 square feet
Therefore, approximately 4,477.44 square feet of cedar are needed to cover the lateral surface area of the octagonal pyramid.
The closest answer provided is:
4,480 ft.²
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