To calculate the total surface area of the square pyramid shaped chocolate dessert, we need to determine the areas of the square base and the four triangular faces.
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Area of the square base: The area \( A \) of a square is given by the formula: \[ A = \text{side}^2 \] For a square with a side length of 3 inches: \[ A = 3^2 = 9 , \text{in}^2 \]
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Area of one triangular face: The area \( A_t \) of a triangle is given by the formula: \[ A_t = \frac{1}{2} \times \text{base} \times \text{height} \] The base of each triangular face is the side length of the square, which is 3 inches, and the height of the triangle is given as 5 inches. Therefore: \[ A_t = \frac{1}{2} \times 3 \times 5 = \frac{15}{2} = 7.5 , \text{in}^2 \]
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Total area of the four triangular faces: Since there are four triangular faces, the total area \( A_{total_triangles} \) of the triangular faces is: \[ A_{total_triangles} = 4 \times A_t = 4 \times 7.5 = 30 , \text{in}^2 \]
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Total surface area of the pyramid: The total surface area \( A_{total} \) of the pyramid is the sum of the area of the base and the total area of the triangular faces: \[ A_{total} = A_{base} + A_{total_triangles} = 9 + 30 = 39 , \text{in}^2 \]
Thus, the total amount of icing needed for the faces of the dessert is 39 in.².