To calculate the surface area of a square pyramid, we need to find the areas of the base and the four triangular faces.
Step 1: Calculate the area of the base
The base of the pyramid is a square, so the area \(A_{\text{base}}\) can be calculated using the formula: \[ A_{\text{base}} = \text{side length}^2 \] Given that the side length of the base is 12: \[ A_{\text{base}} = 12^2 = 144 \]
Step 2: Calculate the area of one triangular face
Each triangular face has a base equal to the side length of the square (12) and a height equal to the slant height (18). The area \(A_{\text{triangle}}\) of one triangular face can be calculated using the formula: \[ A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} \] Substituting the values: \[ A_{\text{triangle}} = \frac{1}{2} \times 12 \times 18 = \frac{1}{2} \times 216 = 108 \]
Step 3: Calculate the area of all four triangular faces
Since there are four triangular faces, the total area for the triangular faces \(A_{\text{triangles}}\) is: \[ A_{\text{triangles}} = 4 \times A_{\text{triangle}} = 4 \times 108 = 432 \]
Step 4: Calculate the total surface area
The total surface area \(A_{\text{total}}\) of the pyramid is the sum of the area of the base and the area of the triangular faces: \[ A_{\text{total}} = A_{\text{base}} + A_{\text{triangles}} = 144 + 432 = 576 \]
Conclusion
The surface area of the square pyramid is: \[ \boxed{576} \]