Question
A triangular prism has an equilateral triangle base whose side length is 6 in. The height of the prism is 10 in. Which of the following correctly calculates the surface area of this triangular prism?(1 point)
Responses
211.18 in.2
211.18 in. squared
304.56 in.2
304.56 in. squared
182.60 in.2
182.60 in. squared
271.14 in.2
Responses
211.18 in.2
211.18 in. squared
304.56 in.2
304.56 in. squared
182.60 in.2
182.60 in. squared
271.14 in.2
Answers
GPT-4o mini
To calculate the surface area of a triangular prism, you need to find the area of the two triangular bases and the area of the three rectangular lateral faces.
1. **Calculate the area of the triangular base:**
The area \( A \) of an equilateral triangle with side length \( s \) can be calculated using the formula:
\[
A = \frac{\sqrt{3}}{4} s^2
\]
For this prism, the side length \( s = 6 \) in:
\[
A = \frac{\sqrt{3}}{4} (6^2) = \frac{\sqrt{3}}{4} \times 36 = 9\sqrt{3} \, \text{in}^2 \approx 15.59 \, \text{in}^2
\]
Since there are two triangular bases:
\[
\text{Total area of the two bases} = 2 \times 9\sqrt{3} \approx 31.18 \, \text{in}^2
\]
2. **Calculate the area of the rectangular lateral faces:**
The lateral faces correspond to the three sides of the triangular base, each with a height equal to the height of the prism \( h = 10 \) in.
- For each side of length \( 6 \) in:
\[
\text{Area of one rectangle} = \text{base} \times \text{height} = 6 \times 10 = 60 \, \text{in}^2
\]
Since there are three rectangular lateral faces:
\[
\text{Total area of the lateral faces} = 3 \times 60 = 180 \, \text{in}^2
\]
3. **Calculate the total surface area:**
\[
\text{Total surface area} = \text{Area of the two bases} + \text{Area of the lateral faces} \approx 31.18 + 180 = 211.18 \, \text{in}^2
\]
Thus, the surface area of the triangular prism is approximately \( 211.18 \, \text{in}^2 \).
The correct answer is:
211.18 in.²
1. **Calculate the area of the triangular base:**
The area \( A \) of an equilateral triangle with side length \( s \) can be calculated using the formula:
\[
A = \frac{\sqrt{3}}{4} s^2
\]
For this prism, the side length \( s = 6 \) in:
\[
A = \frac{\sqrt{3}}{4} (6^2) = \frac{\sqrt{3}}{4} \times 36 = 9\sqrt{3} \, \text{in}^2 \approx 15.59 \, \text{in}^2
\]
Since there are two triangular bases:
\[
\text{Total area of the two bases} = 2 \times 9\sqrt{3} \approx 31.18 \, \text{in}^2
\]
2. **Calculate the area of the rectangular lateral faces:**
The lateral faces correspond to the three sides of the triangular base, each with a height equal to the height of the prism \( h = 10 \) in.
- For each side of length \( 6 \) in:
\[
\text{Area of one rectangle} = \text{base} \times \text{height} = 6 \times 10 = 60 \, \text{in}^2
\]
Since there are three rectangular lateral faces:
\[
\text{Total area of the lateral faces} = 3 \times 60 = 180 \, \text{in}^2
\]
3. **Calculate the total surface area:**
\[
\text{Total surface area} = \text{Area of the two bases} + \text{Area of the lateral faces} \approx 31.18 + 180 = 211.18 \, \text{in}^2
\]
Thus, the surface area of the triangular prism is approximately \( 211.18 \, \text{in}^2 \).
The correct answer is:
211.18 in.²