To find the surface area of a square pyramid, we need to calculate both the area of the base and the area of the four triangular sides (lateral surface area).
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Area of the base: The base of the pyramid is a square. The area \( A_{\text{base}} \) of a square with side length \( a \) is given by: \[ A_{\text{base}} = a^2 \] For a base edge of 1.5 cm: \[ A_{\text{base}} = (1.5 , \text{cm})^2 = 2.25 , \text{cm}^2 \]
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Area of the lateral faces: Each triangular face has a base equal to the edge of the square base and a height equal to the slant height of the pyramid. The area \( A_{\text{triangle}} \) of one triangular face is given by: \[ A_{\text{triangle}} = \frac{1}{2} \times \text{base} \times \text{height} \] The base of each triangular face is the same as the base edge of the pyramid, which is 1.5 cm, and the height is the slant height, which is 4 cm. Hence, for one triangle: \[ A_{\text{triangle}} = \frac{1}{2} \times 1.5 , \text{cm} \times 4 , \text{cm} = \frac{6}{2} , \text{cm}^2 = 3 , \text{cm}^2 \]
Since there are four triangular faces, the total lateral surface area \( A_{\text{lateral}} \) is: \[ A_{\text{lateral}} = 4 \times A_{\text{triangle}} = 4 \times 3 , \text{cm}^2 = 12 , \text{cm}^2 \]
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Total surface area: Now, we combine the area of the base and the lateral surface area to get the total surface area \( A_{\text{total}} \): \[ A_{\text{total}} = A_{\text{base}} + A_{\text{lateral}} = 2.25 , \text{cm}^2 + 12 , \text{cm}^2 = 14.25 , \text{cm}^2 \]
Thus, the surface area of the square pyramid is approximately 14.25 cm² when rounded to the nearest hundredth.