Use the image to answer the question.
An illustration shows a two-dimensional net of a triangular prism with all of its sides open and visible. The net appears as three vertical rectangles placed vertically one on top of the other. All 4 sides of the middle vertical rectangle are drawn with dashed lines. The width of the rectangles is 10 miles. The length of the middle rectangle is 24 miles. Two right triangles adjoin the middle rectangle on the left and right sides, with each base measuring 10 miles and each hypotenuse measuring 26 miles.
Using the net of the triangular prism, what is its surface area?
(1 point)
Responses
1,200 mi.2
, 1,200 mi. squared
840 mi.2
840 mi. squared
720 mi.2
, 720 mi. squared
240 mi.2
, 240 mi. squared
Skip to navigation
The surface area of a triangular prism can be found by adding the areas of its faces. On the net, there are two congruent triangular faces and three rectangular faces. Each triangular face has an area of 1/2 bh, where b is the base and h is the height. In this case, the base is 10 miles and the height can be found using the Pythagorean theorem: h^2 = 26^2 - 10^2, so h = 24 miles. Therefore, each triangular face has an area of 1/2(10)(24) = 120 mi.².
Each rectangular face has an area of lw, where l is the length and w is the width. The top and bottom rectangular faces each have an area of (10)(24) = 240 mi.². The middle rectangular face has two sides with a length of 24 miles and two sides with a length of 26 miles. Therefore, its area can be found using the formula A = 2LH + 2LW + 2HW, where L and H are the longer and shorter sides, respectively, and W is the width. Plugging in the values, we get A = 2(26)(24) + 2(26)(10) + 2(24)(10) = 1,200 mi.².
Adding up the areas of all the faces, we get a total surface area of 120 + 120 + 240 + 240 + 1,200 = 1,920 mi.². Therefore, the correct answer is not listed and should be 1,920 mi.².
Surface Area of Triangular Prisms Quick Check
2 of 52 of 5 Items
Question
Use the image to answer the question.
An illustration shows a two-dimensional net of a triangular prism with all of its sides open and visible. The net appears as three rectangles joined next to each other. All 4 sides of the middle rectangle are drawn with dashed lines. The area of the left rectangle is 48 square inches. The area of the middle rectangle is 64 square inches, and the area of the rectangle on the right is 80 square inches. Two right triangles adjoin the middle rectangle at the top and bottom, with its perpendicular side measuring 6 inches and the common base side marked with a question mark.
Given the following net, what is the area of the triangular base?
(1 point)
Responses
10 inches
10 inches
48 in.2
, 48 in. squared
24 in.2
24 in. squared
8 inches
8 inches
In a triangular prism, the triangular base consists of two congruent triangles. The area of a triangle can be found using the formula 1/2 bh, where b is the base and h is the height. In this case, the perpendicular side of the right triangles is given as 6 inches, and the base of each triangle is one of the sides of the middle rectangle on the net.
Using the Pythagorean theorem, we can find the length of the base of each right triangle. The hypotenuse is the common base of the triangles and is equal to the width of the rectangles, which is not given. Therefore, we cannot directly calculate the area of the triangles.
Since all the other rectangles are given with their areas, we can use the formula for the surface area of a triangular prism to find the area of the triangular base. The surface area of a triangular prism is given by the formula SA = 2B + Ph, where B is the area of the triangular base, P is the perimeter of the base, and h is the height of the prism. The perimeter of the base is the sum of the lengths of its sides, which is twice the length of the middle rectangle plus the unknown base of the right triangles.
The length of the middle rectangle is not given explicitly, but we know that the total area of the middle rectangle is 64 square inches and that its width is the same as that of the other rectangles, which is also unknown. Therefore, we cannot directly calculate the length of the middle rectangle or the perimeter of the triangular base.
Without more information, we cannot find the area of the triangular base. Therefore, the correct answer is not listed and should be "Cannot be determined."
To calculate the surface area of the triangular prism, we need to find the area of each face and then add them together.
1. The base of the prism is a triangle with base 10 miles and height 24 miles. The area of the base is (1/2) * base * height = (1/2) * 10 * 24 = 120 square miles.
2. The top and bottom faces of the prism are rectangles with length 10 miles and width 24 miles. The area of each rectangle is length * width = 10 * 24 = 240 square miles. Since there are two identical faces, the total area for both top and bottom is 2 * 240 = 480 square miles.
3. The two side faces of the prism are rectangles. Each rectangle has a length of 24 miles (the height of the prism) and a width of 10 miles (the length of the base triangle). The area of each side face is 24 * 10 = 240 square miles. Again, since there are two identical side faces, the total area for both sides is 2 * 240 = 480 square miles.
To find the total surface area of the triangular prism, we add the areas of all the faces together:
120 + 480 + 480 = 1,080 square miles.
Therefore, the correct answer is 1,080 mi².