Question
Solve for the surface area of the square pyramid.
(1 point)
Responses
522 ft.2
522 ft. squared
148.5 ft.2
148.5 ft. squared
819 ft.2
819 ft. squared
299.25 ft.2
I got 507 somehow
The area of the base of the square triangle is: 210
The area of one triangular face is: 74.25
(1 point)
Responses
522 ft.2
522 ft. squared
148.5 ft.2
148.5 ft. squared
819 ft.2
819 ft. squared
299.25 ft.2
I got 507 somehow
The area of the base of the square triangle is: 210
The area of one triangular face is: 74.25
Answers
GPT-4o mini
To find the surface area of a square pyramid, you can use the formula:
\[
\text{Surface Area} = \text{Base Area} + \text{Lateral Area}
\]
For a square pyramid:
1. **Base Area**: For a square base with side length \( s \),
\[
\text{Base Area} = s^2
\]
2. **Lateral Area**: The area of the four triangular faces. The area of one triangular face is given by:
\[
\text{Area of one triangular face} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base of each triangle is the same as the side length of the square base, and the height is the slant height of the pyramid.
Given your information:
- The area of the base is \(210 \, \text{ft}^2\).
- The area of one triangular face is \(74.25 \, \text{ft}^2\).
Calculating the lateral area (the area contributed by the four triangular faces):
\[
\text{Lateral Area} = 4 \times \text{Area of one triangular face} = 4 \times 74.25 = 297 \, \text{ft}^2
\]
Now, calculating the total surface area:
\[
\text{Surface Area} = \text{Base Area} + \text{Lateral Area} = 210 + 297 = 507 \, \text{ft}^2
\]
Since you mentioned that you calculated 507, that is indeed the correct surface area.
Regarding the options you provided:
- 522 ft.²
- 148.5 ft.²
- 819 ft.²
- 299.25 ft.²
None of these options match the correct surface area of \(507 \, \text{ft}^2\). Please check if you have all the values right or if there might have been an error in the options given.
\[
\text{Surface Area} = \text{Base Area} + \text{Lateral Area}
\]
For a square pyramid:
1. **Base Area**: For a square base with side length \( s \),
\[
\text{Base Area} = s^2
\]
2. **Lateral Area**: The area of the four triangular faces. The area of one triangular face is given by:
\[
\text{Area of one triangular face} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base of each triangle is the same as the side length of the square base, and the height is the slant height of the pyramid.
Given your information:
- The area of the base is \(210 \, \text{ft}^2\).
- The area of one triangular face is \(74.25 \, \text{ft}^2\).
Calculating the lateral area (the area contributed by the four triangular faces):
\[
\text{Lateral Area} = 4 \times \text{Area of one triangular face} = 4 \times 74.25 = 297 \, \text{ft}^2
\]
Now, calculating the total surface area:
\[
\text{Surface Area} = \text{Base Area} + \text{Lateral Area} = 210 + 297 = 507 \, \text{ft}^2
\]
Since you mentioned that you calculated 507, that is indeed the correct surface area.
Regarding the options you provided:
- 522 ft.²
- 148.5 ft.²
- 819 ft.²
- 299.25 ft.²
None of these options match the correct surface area of \(507 \, \text{ft}^2\). Please check if you have all the values right or if there might have been an error in the options given.