To calculate the surface area of a square pyramid, we need to consider both the area of the base and the area of the four triangular faces.
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Area of the Base: The base of the pyramid is a square. The area \( A_{\text{base}} \) of a square is given by the formula: \[ A_{\text{base}} = \text{side}^2 \] Here, the length of the side of the square base is 12. \[ A_{\text{base}} = 12^2 = 144 \]
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Area of the Triangular Faces: Each triangular face of the pyramid has a base equal to the side length of the square and a height equal to the slant height of the pyramid. The area \( A_{\text{triangle}} \) of one triangular face can be calculated using the formula: \[ A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} \] Here, the base is 12 (the side length of the square) and the height is 18 (the slant height). \[ A_{\text{triangle}} = \frac{1}{2} \times 12 \times 18 = \frac{1}{2} \times 216 = 108 \]
Since there are 4 triangular faces, the total area of the triangular faces \( A_{\text{triangular faces}} \) is: \[ A_{\text{triangular faces}} = 4 \times A_{\text{triangle}} = 4 \times 108 = 432 \]
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Total Surface Area: Now, we can calculate the total surface area \( A_{\text{total}} \) of the pyramid by adding the area of the base and the area of the triangular faces: \[ A_{\text{total}} = A_{\text{base}} + A_{\text{triangular faces}} \] \[ A_{\text{total}} = 144 + 432 = 576 \]
Thus, the total surface area of the square pyramid is \( \boxed{576} \).