To find the surface area of Sally's square pyramid, we need to consider both the base and the lateral faces.
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Base area: The base is a square with side length of 4 cm. The area of the base is: \[ \text{Area}_{\text{base}} = \text{side}^2 = 4 , \text{cm} \times 4 , \text{cm} = 16 , \text{cm}^2 \]
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Lateral area: A square pyramid has four triangular lateral faces. The area of one triangular face is given by: \[ \text{Area}_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} \] In this case, the base of each triangle is equal to the side of the square base (4 cm), and the height of each triangle is the slant height. The problem states that the length of the pyramid is 5 cm, which we will assume is the slant height.
So, the area of one triangular face would be: \[ \text{Area}_{\text{triangle}} = \frac{1}{2} \times 4 , \text{cm} \times 5 , \text{cm} = \frac{20}{2} = 10 , \text{cm}^2 \]
Thus, the total lateral area (for 4 triangular faces) is: \[ \text{Lateral Area} = 4 \times \text{Area}_{\text{triangle}} = 4 \times 10 , \text{cm}^2 = 40 , \text{cm}^2 \]
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Total surface area: Now, we add the area of the base and the lateral area together: \[ \text{Surface Area} = \text{Area}_{\text{base}} + \text{Lateral Area} = 16 , \text{cm}^2 + 40 , \text{cm}^2 = 56 , \text{cm}^2 \]
Therefore, the surface area of Sally's game piece is 56 cm².