To calculate the surface area of a square pyramid, we need to find 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 of a square is calculated using the formula: \[ \text{Area}{\text{base}} = \text{side} \times \text{side} \] Given that the base length (side) is 12: \[ \text{Area}{\text{base}} = 12 \times 12 = 144 \text{ square units} \]
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Area of the Triangular Faces: Each triangular face has a base equal to the side length of the square, which is 12, and a height equal to the slant height of the pyramid, which is 18. The area of one triangle is calculated using the formula: \[ \text{Area}{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} \] For one triangular face: \[ \text{Area}{\text{triangle}} = \frac{1}{2} \times 12 \times 18 \] \[ \text{Area}_{\text{triangle}} = \frac{1}{2} \times 216 = 108 \text{ square units} \]
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Total Area of the Four Triangular Faces: Since there are four triangular faces, we multiply the area of one triangle by 4: \[ \text{Total Area}_{\text{triangles}} = 4 \times 108 = 432 \text{ square units} \]
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Total Surface Area of the Pyramid: Finally, we add the area of the base to the total area of the triangular faces: \[ \text{Total Surface Area} = \text{Area}{\text{base}} + \text{Total Area}{\text{triangles}} \] \[ \text{Total Surface Area} = 144 + 432 = 576 \text{ square units} \]
Thus, the surface area of the square pyramid is: \[ \boxed{576 \text{ square units}} \]