To calculate the surface area of a rectangular pyramid, we need to consider both the area of the base and the area of the triangular faces.
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Calculate the area of the rectangular base: \[ \text{Area of the base} = \text{length} \times \text{width} = 14 , \text{cm} \times 12 , \text{cm} = 168 , \text{cm}^2 \]
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Calculate the area of the triangular faces: The pyramid has two triangular faces corresponding to the length and width of the base.
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For the triangular faces with base length 14 cm: The height from the apex to the base of these triangles is 10.6 cm. \[ \text{Area of one triangular face} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 14 , \text{cm} \times 10.6 , \text{cm} = 74.2 , \text{cm}^2 \] Since there are two such triangular faces: \[ \text{Total area for two triangular faces} = 2 \times 74.2 , \text{cm}^2 = 148.4 , \text{cm}^2 \]
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For the triangular faces with base width 12 cm: The height from the apex to the base of these triangles is 10 cm. \[ \text{Area of one triangular face} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12 , \text{cm} \times 10 , \text{cm} = 60 , \text{cm}^2 \] Since there are two such triangular faces: \[ \text{Total area for two triangular faces} = 2 \times 60 , \text{cm}^2 = 120 , \text{cm}^2 \]
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Combine all areas to find the total surface area: \[ \text{Total Surface Area} = \text{Area of the base} + \text{Area of two triangular faces (length)} + \text{Area of two triangular faces (width)} \] \[ \text{Total Surface Area} = 168 , \text{cm}^2 + 148.4 , \text{cm}^2 + 120 , \text{cm}^2 \] \[ \text{Total Surface Area} = 436.4 , \text{cm}^2 \]
Thus, the surface area of the rectangular pyramid is \( \text{436.4 cm}^2 \).