To find the surface area of a square pyramid that includes the base, we'll calculate the area of the base and the area of the four triangular faces.
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Calculate the area of the base: The base is a square with a side length of \(10\) feet. The area \((A_{base})\) of the base can be calculated using the formula for the area of a square:
\[ A_{base} = \text{side}^2 = 10 \times 10 = 100 \text{ ft}^2 \]
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Calculate the area of one triangular face: Each triangular face has a base equal to the side of the square, which is \(10\) feet, and a height equal to \(12\) feet. The area \((A_{triangle})\) of one triangle can be calculated using the formula for the area of a triangle:
\[ A_{triangle} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 \times 12 = 60 \text{ ft}^2 \]
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Calculate the total area of the four triangular faces: Since there are four triangular faces, we multiply the area of one triangle by \(4\):
\[ A_{triangles\ total} = 4 \times A_{triangle} = 4 \times 60 = 240 \text{ ft}^2 \]
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Calculate the total surface area of the pyramid: The total surface area is the sum of the area of the base and the total area of the triangular faces:
\[ A_{total} = A_{base} + A_{triangles\ total} = 100 + 240 = 340 \text{ ft}^2 \]
Thus, the surface area of the garden sculpture, including the base, is
\[ \boxed{340} \text{ ft}^2. \]