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:
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Base Area: For a square base with side length \( s \), \[ \text{Base Area} = s^2 \]
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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.