Question
Right Rectangular Prisms & Pyramids Online Practice
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Question
Find the surface area of a triangular pyramid with base 10, a base height of 8.7, and a slant height of 12.(1 point)
Responses
103.5
103.5
216.6
216.6
223.5
223.5
232.2
Answers
GPT-4o mini
To find the surface area of a triangular pyramid, you need to calculate the area of the base and the area of the three triangular faces and then sum them together.
1. **Area of the base** (which is a triangle):
- The formula for the area of a triangle is \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \).
- Given the base of the triangle is 10 and the height is 8.7:
\[
\text{Area}_{\text{base}} = \frac{1}{2} \times 10 \times 8.7 = 43.5
\]
2. **Area of the triangular faces**:
- Each triangular face has a base equal to the side of the base triangle (which is 10) and a height equal to the slant height (12). There are three triangular faces.
- The area of one triangular face would be:
\[
\text{Area}_{\text{face}} = \frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 10 \times 12 = 60
\]
- Since there are three faces, the total area of the triangular faces is:
\[
\text{Total Area}_{\text{faces}} = 3 \times 60 = 180
\]
3. **Total Surface Area**:
- Now, sum the area of the base and the total area of the triangular faces:
\[
\text{Total Surface Area} = \text{Area}_{\text{base}} + \text{Total Area}_{\text{faces}} = 43.5 + 180 = 223.5
\]
Therefore, the surface area of the triangular pyramid is **223.5**.
So, the correct response is: **223.5**.
1. **Area of the base** (which is a triangle):
- The formula for the area of a triangle is \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \).
- Given the base of the triangle is 10 and the height is 8.7:
\[
\text{Area}_{\text{base}} = \frac{1}{2} \times 10 \times 8.7 = 43.5
\]
2. **Area of the triangular faces**:
- Each triangular face has a base equal to the side of the base triangle (which is 10) and a height equal to the slant height (12). There are three triangular faces.
- The area of one triangular face would be:
\[
\text{Area}_{\text{face}} = \frac{1}{2} \times \text{base} \times \text{slant height} = \frac{1}{2} \times 10 \times 12 = 60
\]
- Since there are three faces, the total area of the triangular faces is:
\[
\text{Total Area}_{\text{faces}} = 3 \times 60 = 180
\]
3. **Total Surface Area**:
- Now, sum the area of the base and the total area of the triangular faces:
\[
\text{Total Surface Area} = \text{Area}_{\text{base}} + \text{Total Area}_{\text{faces}} = 43.5 + 180 = 223.5
\]
Therefore, the surface area of the triangular pyramid is **223.5**.
So, the correct response is: **223.5**.