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?
Answers
GPT-4o mini
To find the surface area of a triangular prism, we need to calculate the area of the triangular bases and the area of the three rectangular faces.
1. **Area of the triangular base:**
The base is an equilateral triangle with side length \( s = 6 \) inches. The area \( A \) of an equilateral triangle can be calculated using the formula:
\[
A = \frac{\sqrt{3}}{4} s^2
\]
Plugging in the side length:
\[
A = \frac{\sqrt{3}}{4} (6)^2 = \frac{\sqrt{3}}{4} \cdot 36 = 9\sqrt{3} \text{ square inches}
\]
2. **Area of the two triangular bases:**
Since there are two triangular bases, the total area of the triangular bases is:
\[
2A = 2 \times 9\sqrt{3} = 18\sqrt{3} \text{ square inches}
\]
3. **Area of the rectangular faces:**
The prism has three rectangular sides, and we have to calculate their areas:
- The height of the prism is \( h = 10 \) inches.
- The lengths of the sides of the triangular base are all \( 6 \) inches.
The areas of the three rectangular faces are calculated as follows:
- First rectangle (base side 6 inches):
\[
\text{Area}_1 = \text{Base side} \times \text{Height} = 6 \times 10 = 60 \text{ square inches}
\]
- Second rectangle (base side 6 inches):
\[
\text{Area}_2 = 6 \times 10 = 60 \text{ square inches}
\]
- Third rectangle (base side 6 inches):
\[
\text{Area}_3 = 6 \times 10 = 60 \text{ square inches}
\]
Thus, the total area of the rectangular faces is:
\[
\text{Total Rectangular Area} = 60 + 60 + 60 = 180 \text{ square inches}
\]
4. **Total surface area of the prism:**
Finally, the total surface area \( SA \) of the triangular prism is the sum of the area of the triangular bases and the area of the rectangular faces:
\[
SA = \text{Area of the triangular bases} + \text{Area of the rectangular faces}
\]
\[
SA = 18\sqrt{3} + 180
\]
So, the surface area of the triangular prism is:
\[
SA = 18\sqrt{3} + 180 \text{ square inches}
\]
1. **Area of the triangular base:**
The base is an equilateral triangle with side length \( s = 6 \) inches. The area \( A \) of an equilateral triangle can be calculated using the formula:
\[
A = \frac{\sqrt{3}}{4} s^2
\]
Plugging in the side length:
\[
A = \frac{\sqrt{3}}{4} (6)^2 = \frac{\sqrt{3}}{4} \cdot 36 = 9\sqrt{3} \text{ square inches}
\]
2. **Area of the two triangular bases:**
Since there are two triangular bases, the total area of the triangular bases is:
\[
2A = 2 \times 9\sqrt{3} = 18\sqrt{3} \text{ square inches}
\]
3. **Area of the rectangular faces:**
The prism has three rectangular sides, and we have to calculate their areas:
- The height of the prism is \( h = 10 \) inches.
- The lengths of the sides of the triangular base are all \( 6 \) inches.
The areas of the three rectangular faces are calculated as follows:
- First rectangle (base side 6 inches):
\[
\text{Area}_1 = \text{Base side} \times \text{Height} = 6 \times 10 = 60 \text{ square inches}
\]
- Second rectangle (base side 6 inches):
\[
\text{Area}_2 = 6 \times 10 = 60 \text{ square inches}
\]
- Third rectangle (base side 6 inches):
\[
\text{Area}_3 = 6 \times 10 = 60 \text{ square inches}
\]
Thus, the total area of the rectangular faces is:
\[
\text{Total Rectangular Area} = 60 + 60 + 60 = 180 \text{ square inches}
\]
4. **Total surface area of the prism:**
Finally, the total surface area \( SA \) of the triangular prism is the sum of the area of the triangular bases and the area of the rectangular faces:
\[
SA = \text{Area of the triangular bases} + \text{Area of the rectangular faces}
\]
\[
SA = 18\sqrt{3} + 180
\]
So, the surface area of the triangular prism is:
\[
SA = 18\sqrt{3} + 180 \text{ square inches}
\]