Asked by pls noooo
Use the image to answer the question.
An illustration shows a triangular prism. The front triangular face is visible. The right rectangular face is visible. The visible rectangular face has a width of 5 centimeters and length of 11 centimeters. The sides and edges that are not visible are indicated by dashed lines. The legs of the triangular face are labeled 3 centimeters and 4 centimeters. The hypotenuse shares the same label as the width of the visible rectangular face.
What is the surface area of the triangular prism?
(1 point)
__square centimeters
An illustration shows a triangular prism. The front triangular face is visible. The right rectangular face is visible. The visible rectangular face has a width of 5 centimeters and length of 11 centimeters. The sides and edges that are not visible are indicated by dashed lines. The legs of the triangular face are labeled 3 centimeters and 4 centimeters. The hypotenuse shares the same label as the width of the visible rectangular face.
What is the surface area of the triangular prism?
(1 point)
__square centimeters
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To calculate the surface area of the triangular prism, we first need to determine the area of the triangular base and then the area of the rectangular faces.
### Step 1: Calculate the area of the triangular base
The triangular face is a right triangle with legs measuring 3 cm and 4 cm. The area \( A \) of a triangle is given by the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
In this case, we can use the legs of the triangle as the base and height. So,
\[
A = \frac{1}{2} \times 3 \, \text{cm} \times 4 \, \text{cm} = \frac{1}{2} \times 12 \, \text{cm}^2 = 6 \, \text{cm}^2
\]
### Step 2: Calculate the area of the rectangular faces
The prism has three rectangular faces: two of the same size corresponding to the triangular base and one rectangular face given in the problem.
1. **Area of the rectangular faces corresponding to the triangle:**
The height of the prism is the same as the hypotenuse of the triangle, which can be calculated using the Pythagorean theorem:
\[
\text{hypotenuse} = \sqrt{(3 \, \text{cm})^2 + (4 \, \text{cm})^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \, \text{cm}
\]
The two rectangular faces corresponding to the legs of the triangle:
- For the side of length 3 cm:
\[
\text{Area} = \text{leg} \times \text{height} = 3 \, \text{cm} \times 11 \, \text{cm} = 33 \, \text{cm}^2
\]
- For the side of length 4 cm:
\[
\text{Area} = \text{leg} \times \text{height} = 4 \, \text{cm} \times 11 \, \text{cm} = 44 \, \text{cm}^2
\]
2. **Area of the base rectangle:**
The width of this rectangular face is given as 5 cm, and its height is 11 cm. Thus,
\[
\text{Area} = \text{width} \times \text{height} = 5 \, \text{cm} \times 11 \, \text{cm} = 55 \, \text{cm}^2
\]
### Step 3: Total surface area calculation
Add all the areas together to find the total surface area of the prism:
\[
\text{Total Surface Area} = \text{Area of triangular base} + 2(\text{Area corresponding to legs}) + \text{Area of base triangle}
\]
\[
= 6 \, \text{cm}^2 + 33 \, \text{cm}^2 + 44 \, \text{cm}^2 + 55 \, \text{cm}^2
\]
\[
= 6 + 33 + 44 + 55
= 138 \, \text{cm}^2
\]
Thus, the surface area of the triangular prism is:
\[
\boxed{138} \text{ square centimeters}
\]
### Step 1: Calculate the area of the triangular base
The triangular face is a right triangle with legs measuring 3 cm and 4 cm. The area \( A \) of a triangle is given by the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
In this case, we can use the legs of the triangle as the base and height. So,
\[
A = \frac{1}{2} \times 3 \, \text{cm} \times 4 \, \text{cm} = \frac{1}{2} \times 12 \, \text{cm}^2 = 6 \, \text{cm}^2
\]
### Step 2: Calculate the area of the rectangular faces
The prism has three rectangular faces: two of the same size corresponding to the triangular base and one rectangular face given in the problem.
1. **Area of the rectangular faces corresponding to the triangle:**
The height of the prism is the same as the hypotenuse of the triangle, which can be calculated using the Pythagorean theorem:
\[
\text{hypotenuse} = \sqrt{(3 \, \text{cm})^2 + (4 \, \text{cm})^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \, \text{cm}
\]
The two rectangular faces corresponding to the legs of the triangle:
- For the side of length 3 cm:
\[
\text{Area} = \text{leg} \times \text{height} = 3 \, \text{cm} \times 11 \, \text{cm} = 33 \, \text{cm}^2
\]
- For the side of length 4 cm:
\[
\text{Area} = \text{leg} \times \text{height} = 4 \, \text{cm} \times 11 \, \text{cm} = 44 \, \text{cm}^2
\]
2. **Area of the base rectangle:**
The width of this rectangular face is given as 5 cm, and its height is 11 cm. Thus,
\[
\text{Area} = \text{width} \times \text{height} = 5 \, \text{cm} \times 11 \, \text{cm} = 55 \, \text{cm}^2
\]
### Step 3: Total surface area calculation
Add all the areas together to find the total surface area of the prism:
\[
\text{Total Surface Area} = \text{Area of triangular base} + 2(\text{Area corresponding to legs}) + \text{Area of base triangle}
\]
\[
= 6 \, \text{cm}^2 + 33 \, \text{cm}^2 + 44 \, \text{cm}^2 + 55 \, \text{cm}^2
\]
\[
= 6 + 33 + 44 + 55
= 138 \, \text{cm}^2
\]
Thus, the surface area of the triangular prism is:
\[
\boxed{138} \text{ square centimeters}
\]
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