Question
Triangular prism
height 4 cm
Side 5 cm
Width 6 cm
Length 15 cm
What is the surface area of this triangular prism.
240 cm²
288 cm²
140 cm²
264 cm²
height 4 cm
Side 5 cm
Width 6 cm
Length 15 cm
What is the surface area of this triangular prism.
240 cm²
288 cm²
140 cm²
264 cm²
Answers
Answered by
GPT-4o mini
To calculate the surface area of a triangular prism, we can use the following formula:
\[
\text{Surface Area} = \text{Base Area} + \text{Lateral Area}
\]
1. **Base Area (A)** of the triangular face (which is a triangle):
The area \( A \) of a triangle can be calculated with the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
In this case, assuming "side" is the base of the triangle (5 cm) and "height" is the height of the triangle:
\[
A = \frac{1}{2} \times 5 \, \text{cm} \times 4 \, \text{cm} = \frac{1}{2} \times 20 \, \text{cm}^2 = 10 \, \text{cm}^2
\]
Since the triangular prism has two triangular bases, the total area for both bases is:
\[
\text{Base Area} = 2A = 2 \times 10 \, \text{cm}^2 = 20 \, \text{cm}^2
\]
2. **Lateral Area (LA)**:
The lateral area consists of the three rectangular faces. To find the lateral area, calculate the area of each rectangle. The dimensions of the rectangles are:
- Rectangle between the height and side lengths (height of the prism)
- Rectangle for the base of the triangle (base of the triangle and length of the prism)
- Rectangle for the other side of the triangle (the side length of the triangle and length of the prism)
The lateral faces are:
- One rectangle with dimensions \( (5 \, \text{cm} \times 15 \, \text{cm}) \)
- One rectangle with dimensions \( (6 \, \text{cm} \times 15 \, \text{cm}) \)
- One rectangle with dimensions \( (5 \, \text{cm} \times 15 \, \text{cm}) \)
Therefore, the areas of the rectangles are:
- Area of rectangle 1:
\[
5 \, \text{cm} \times 15 \, \text{cm} = 75 \, \text{cm}^2
\]
- Area of rectangle 2:
\[
6 \, \text{cm} \times 15 \, \text{cm} = 90 \, \text{cm}^2
\]
- Area of rectangle 3:
\[
5 \, \text{cm} \times 15 \, \text{cm} = 75 \, \text{cm}^2
\]
Summing the area of the lateral faces gives:
\[
LA = 75 \, \text{cm}^2 + 90 \, \text{cm}^2 + 75 \, \text{cm}^2 = 240 \, \text{cm}^2
\]
3. **Total Surface Area**:
Now we can calculate the total surface area:
\[
\text{Total Surface Area} = \text{Base Area} + \text{Lateral Area} = 20 \, \text{cm}^2 + 240 \, \text{cm}^2 = 260 \, \text{cm}^2
\]
After reviewing the options provided, it appears none of the answer choices are correct for the calculations for surface area based on the dimensions provided.
Re-evaluating the calculations and options might be necessary, or adjustments may need to be made depending on what interpretations were made for dimensions not specified clearly.
Ensure to clarify the dimensions again:
1. Which side refers to the base?
2. The correct lengths for lateral face calculations according to the dimensions you are using.
For the options given, re-checking every area calculation for accuracy could be warranted if the question's parameters change. If you still seek clarity or specific adjustments to this prism case, let me know!
\[
\text{Surface Area} = \text{Base Area} + \text{Lateral Area}
\]
1. **Base Area (A)** of the triangular face (which is a triangle):
The area \( A \) of a triangle can be calculated with the formula:
\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\]
In this case, assuming "side" is the base of the triangle (5 cm) and "height" is the height of the triangle:
\[
A = \frac{1}{2} \times 5 \, \text{cm} \times 4 \, \text{cm} = \frac{1}{2} \times 20 \, \text{cm}^2 = 10 \, \text{cm}^2
\]
Since the triangular prism has two triangular bases, the total area for both bases is:
\[
\text{Base Area} = 2A = 2 \times 10 \, \text{cm}^2 = 20 \, \text{cm}^2
\]
2. **Lateral Area (LA)**:
The lateral area consists of the three rectangular faces. To find the lateral area, calculate the area of each rectangle. The dimensions of the rectangles are:
- Rectangle between the height and side lengths (height of the prism)
- Rectangle for the base of the triangle (base of the triangle and length of the prism)
- Rectangle for the other side of the triangle (the side length of the triangle and length of the prism)
The lateral faces are:
- One rectangle with dimensions \( (5 \, \text{cm} \times 15 \, \text{cm}) \)
- One rectangle with dimensions \( (6 \, \text{cm} \times 15 \, \text{cm}) \)
- One rectangle with dimensions \( (5 \, \text{cm} \times 15 \, \text{cm}) \)
Therefore, the areas of the rectangles are:
- Area of rectangle 1:
\[
5 \, \text{cm} \times 15 \, \text{cm} = 75 \, \text{cm}^2
\]
- Area of rectangle 2:
\[
6 \, \text{cm} \times 15 \, \text{cm} = 90 \, \text{cm}^2
\]
- Area of rectangle 3:
\[
5 \, \text{cm} \times 15 \, \text{cm} = 75 \, \text{cm}^2
\]
Summing the area of the lateral faces gives:
\[
LA = 75 \, \text{cm}^2 + 90 \, \text{cm}^2 + 75 \, \text{cm}^2 = 240 \, \text{cm}^2
\]
3. **Total Surface Area**:
Now we can calculate the total surface area:
\[
\text{Total Surface Area} = \text{Base Area} + \text{Lateral Area} = 20 \, \text{cm}^2 + 240 \, \text{cm}^2 = 260 \, \text{cm}^2
\]
After reviewing the options provided, it appears none of the answer choices are correct for the calculations for surface area based on the dimensions provided.
Re-evaluating the calculations and options might be necessary, or adjustments may need to be made depending on what interpretations were made for dimensions not specified clearly.
Ensure to clarify the dimensions again:
1. Which side refers to the base?
2. The correct lengths for lateral face calculations according to the dimensions you are using.
For the options given, re-checking every area calculation for accuracy could be warranted if the question's parameters change. If you still seek clarity or specific adjustments to this prism case, let me know!