To calculate the surface area of a square pyramid, we need to consider both the area of the base and the area of the triangular faces.
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Area of the base: The base of the square pyramid is a square. The area \( A_{base} \) of the base can be calculated using the formula: \[ A_{base} = \text{side}^2 \] For a base length of 12: \[ A_{base} = 12^2 = 144 \]
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Area of the triangular faces: A square pyramid has 4 triangular faces. The area \( A_{triangle} \) of one triangular face can be calculated using the formula: \[ A_{triangle} = \frac{1}{2} \times \text{base} \times \text{height} \] In this case, the base of the triangle is equal to the side of the square base, which is 12. The height of the triangle is given by the slant height of the pyramid, which is 18.
Thus, the area of one triangular face is: \[ A_{triangle} = \frac{1}{2} \times 12 \times 18 \] Calculating that: \[ A_{triangle} = \frac{1}{2} \times 12 \times 18 = \frac{1}{2} \times 216 = 108 \]
Since there are 4 triangular faces, the total area of the triangular faces \( A_{triangles} \) is: \[ A_{triangles} = 4 \times A_{triangle} = 4 \times 108 = 432 \]
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Total surface area: The total surface area \( A_{total} \) of the pyramid is the sum of the area of the base and the area of the triangular faces: \[ A_{total} = A_{base} + A_{triangles} = 144 + 432 \] Therefore: \[ A_{total} = 576 \]
Thus, the surface area of the square pyramid is 576 square units.