Question
Show your steps and explain how to find the surface area of the triangular prism below.
(4 points)
(4 points)
Answers
Answered by
GPT-4o mini
To find the surface area of a triangular prism, you need to calculate the areas of all its faces and sum them up. Here are the steps to find the surface area:
### Step 1: Identify Dimensions
First, identify the dimensions of the prism, including the base and height of the triangular face and the length of the prism. Let’s assume the following dimensions for an example:
- Base of the triangle (b)
- Height of the triangle (h)
- Length of the prism (L)
### Step 2: Calculate the Area of the Triangular Face
The area (A_triangle) of a triangular face can be calculated using the formula:
\[
A_{\text{triangle}} = \frac{1}{2} \times b \times h
\]
### Step 3: Calculate the Area of the Rectangular Faces
The triangular prism has three rectangular lateral faces. The areas of these rectangles can be calculated as follows:
1. **Rectangle corresponding to the base of the triangle:**
\[
A_{\text{rect1}} = b \times L
\]
2. **Rectangle corresponding to one of the other sides of the triangle:**
Let's denote the second side of the triangle as \( s_2 \).
\[
A_{\text{rect2}} = s_2 \times L
\]
3. **Rectangle corresponding to the third side of the triangle:**
Let’s denote the third side of the triangle as \( s_3 \).
\[
A_{\text{rect3}} = s_3 \times L
\]
### Step 4: Sum the Areas
Now, you can find the total surface area (SA) of the triangular prism by adding the areas of the two triangular faces and the three rectangular faces:
\[
SA = 2 \times A_{\text{triangle}} + A_{\text{rect1}} + A_{\text{rect2}} + A_{\text{rect3}}
\]
### Example
Let's use hypothetical values:
- Base \( b = 5 \, \text{units} \)
- Height \( h = 3 \, \text{units} \)
- Length \( L = 10 \, \text{units} \)
- Sides of the triangle (assuming they are \( s_2 = 4 \) units and \( s_3 = 6 \) units)
1. **Calculate the area of the triangular face:**
\[
A_{\text{triangle}} = \frac{1}{2} \times 5 \times 3 = 7.5 \, \text{square units}
\]
2. **Calculate the areas of the rectangular faces:**
\[
A_{\text{rect1}} = 5 \times 10 = 50 \, \text{square units}
\]
\[
A_{\text{rect2}} = 4 \times 10 = 40 \, \text{square units}
\]
\[
A_{\text{rect3}} = 6 \times 10 = 60 \, \text{square units}
\]
3. **Now, calculate the total surface area:**
\[
SA = 2 \times 7.5 + 50 + 40 + 60 = 15 + 50 + 40 + 60 = 165 \, \text{square units}
\]
### Conclusion
The surface area of the triangular prism is **165 square units**. You can replace the dimensions and repeat the steps to find the surface area for any specific triangular prism.
### Step 1: Identify Dimensions
First, identify the dimensions of the prism, including the base and height of the triangular face and the length of the prism. Let’s assume the following dimensions for an example:
- Base of the triangle (b)
- Height of the triangle (h)
- Length of the prism (L)
### Step 2: Calculate the Area of the Triangular Face
The area (A_triangle) of a triangular face can be calculated using the formula:
\[
A_{\text{triangle}} = \frac{1}{2} \times b \times h
\]
### Step 3: Calculate the Area of the Rectangular Faces
The triangular prism has three rectangular lateral faces. The areas of these rectangles can be calculated as follows:
1. **Rectangle corresponding to the base of the triangle:**
\[
A_{\text{rect1}} = b \times L
\]
2. **Rectangle corresponding to one of the other sides of the triangle:**
Let's denote the second side of the triangle as \( s_2 \).
\[
A_{\text{rect2}} = s_2 \times L
\]
3. **Rectangle corresponding to the third side of the triangle:**
Let’s denote the third side of the triangle as \( s_3 \).
\[
A_{\text{rect3}} = s_3 \times L
\]
### Step 4: Sum the Areas
Now, you can find the total surface area (SA) of the triangular prism by adding the areas of the two triangular faces and the three rectangular faces:
\[
SA = 2 \times A_{\text{triangle}} + A_{\text{rect1}} + A_{\text{rect2}} + A_{\text{rect3}}
\]
### Example
Let's use hypothetical values:
- Base \( b = 5 \, \text{units} \)
- Height \( h = 3 \, \text{units} \)
- Length \( L = 10 \, \text{units} \)
- Sides of the triangle (assuming they are \( s_2 = 4 \) units and \( s_3 = 6 \) units)
1. **Calculate the area of the triangular face:**
\[
A_{\text{triangle}} = \frac{1}{2} \times 5 \times 3 = 7.5 \, \text{square units}
\]
2. **Calculate the areas of the rectangular faces:**
\[
A_{\text{rect1}} = 5 \times 10 = 50 \, \text{square units}
\]
\[
A_{\text{rect2}} = 4 \times 10 = 40 \, \text{square units}
\]
\[
A_{\text{rect3}} = 6 \times 10 = 60 \, \text{square units}
\]
3. **Now, calculate the total surface area:**
\[
SA = 2 \times 7.5 + 50 + 40 + 60 = 15 + 50 + 40 + 60 = 165 \, \text{square units}
\]
### Conclusion
The surface area of the triangular prism is **165 square units**. You can replace the dimensions and repeat the steps to find the surface area for any specific triangular prism.