Use the image to answer the question.
An illustration shows a triangular pyramid with an equilateral base. The perpendicular height of the base measures 12 inches and all sides measure 14 inches. The slant height measures 13 inches.
Use nets to find the surface area of the triangular pyramid with an equilateral base.
(1 point)
Responses
175 in.2
175 in. squared
357 in.2
357 in. squared
84 in.2
84 in. squared
273 in.2
273 in. squared
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To find the surface area of the triangular pyramid, we need to find the area of the base and the area of the three triangular faces.
The area of an equilateral triangle base can be calculated using the formula Area = (sqrt(3)/4) * s^2, where s is the side length of the equilateral triangle. In this case, s = 14, so the area of the base is (sqrt(3)/4) * 14^2 = 84 in^2.
The area of each triangular face can be calculated by using the formula Area = 0.5 * base * height. The height of each triangular face is the slant height, which is 13 inches, and the base of each triangular face is the side length of the base, 14 inches.
So the area of each triangular face is 0.5 * 14 * 13 = 91 in^2.
Since there are three triangular faces, the total area of the three faces is 3 * 91 = 273 in^2.
Therefore, the total surface area of the triangular pyramid is the sum of the base area and the three triangular faces, which is 84 + 273 = 357 in^2.
Therefore, the correct answer is:
357 in.2
357 in. squared
Use the image to answer the question.
An illustration shows a net diagram of a tetrahedron. The net diagram is in the shape of a parallelogram. The net diagram is formed by connecting 4 identical triangles horizontally. From left to right, the first and third triangles rest on their bases. The second and fourth triangles are inverted. The side edges where each triangle meets the next are marked by dashed lines.
The tetrahedron has a surface area of 20 m2 . What is the area of one face of the tetrahedron?
(1 point)
Responses
15 m2
15 m squared
10 m2
10 m squared
20 m2
20 m squared
5 m2
To find the area of one face of the tetrahedron, we first need to determine how many faces the tetrahedron has. Based on the net diagram described, the tetrahedron is made up of 4 identical triangles. Therefore, it has 4 faces.
Since the surface area of the tetrahedron is 20 m^2, and it is made up of 4 faces, each face would have an area of: 20 m^2 / 4 = 5 m^2.
Therefore, the area of one face of the tetrahedron is:
5 m^2
Use the image to answer the question.
An illustration shows a net diagram of a tetrahedron. The net diagram is in the shape of a parallelogram. The net diagram is formed by connecting 4 identical triangles horizontally. From left to right, the first and third triangles rest on their bases. The second and fourth triangles are inverted. The side edges where each triangle meets the next are marked by dashed lines. The base length of the first triangle measures 8 inches. The slant height is indicated by a perpendicular line on the first triangle and measures 7 inches.
What is the surface area of the tetrahedron?
(1 point)
Responses
224 in.2
224 in. squared
112 in.2
112 in. squared
84 in.2
84 in. squared
28 in.2
28 in. squared
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To find the surface area of the tetrahedron, we first need to find the area of the base triangle and then calculate the lateral area of the three identical triangles that form the sides of the tetrahedron.
The area of the base triangle can be calculated using the formula Area = 0.5 * base * height, where the base is 8 inches and the height (slant height) is 7 inches.
Area of base triangle = 0.5 * 8 * 7 = 28 square inches.
Since there are 4 identical triangles in the net diagram, one as the base and three as the sides, the total lateral area of the tetrahedron is 3 times the area of one triangle. Therefore, the lateral area is 3 * 28 = 84 square inches.
The total surface area of the tetrahedron is the sum of the base area and the lateral area, which is 28 + 84 = 112 square inches.
Therefore, the surface area of the tetrahedron is 112 in^2.
So, the correct response is:
112 in.2
112 in. squared