To find the surface area of a square pyramid, we can use the formula:
\[ \text{Surface Area} = \text{Base Area} + \text{Lateral Area} \]
- Base Area: For a square pyramid, the base area (A) is given by \( A = \text{side}^2 \). Since the base edge is 8 ft:
\[ \text{Base Area} = 8 , \text{ft} \times 8 , \text{ft} = 64 , \text{ft}^2 \]
- Lateral Area: The lateral area (LA) of a pyramid is the sum of the areas of the triangular faces. For a square pyramid, there are 4 triangular faces, and the area \( A_t \) of one triangular face can be calculated as:
\[ A_t = \frac{1}{2} \times \text{base} \times \text{slant height} \] Here, the base of each triangle is the same as the base edge of the pyramid (8 ft), and the slant height is given as 16 ft:
\[ A_t = \frac{1}{2} \times 8 , \text{ft} \times 16 , \text{ft} = \frac{1}{2} \times 128 , \text{ft}^2 = 64 , \text{ft}^2 \]
Therefore, the total lateral area for all 4 triangular faces is:
\[ \text{Lateral Area} = 4 \times A_t = 4 \times 64 , \text{ft}^2 = 256 , \text{ft}^2 \]
- Total Surface Area: Now, we sum the base area and the lateral area:
\[ \text{Surface Area} = \text{Base Area} + \text{Lateral Area} = 64 , \text{ft}^2 + 256 , \text{ft}^2 = 320 , \text{ft}^2 \]
Thus, the total surface area of the square pyramid is:
\[ \boxed{320 , \text{ft}^2} \]