To find the surface area of a rectangular pyramid, we need to calculate the area of the base and the areas of the triangular lateral faces.
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Calculate the area of the base: The base is a rectangle, so the area \( A_{base} \) is given by: \[ A_{base} = \text{length} \times \text{width} = 14 , \text{cm} \times 12 , \text{cm} = 168 , \text{cm}^2 \]
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Calculate the area of the lateral triangular faces: There are four triangular faces. We will calculate the area of each triangle:
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Area of the two triangular faces with base 14 cm: For these triangles, the height from the apex to the base of 14 cm is 10 cm. The area \( A_{triangle1} \) of one triangle is: \[ A_{triangle1} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 14 , \text{cm} \times 10 , \text{cm} = 70 , \text{cm}^2 \] Since there are two such triangles, their total area will be: \[ A_{triangles1} = 2 \times A_{triangle1} = 2 \times 70 , \text{cm}^2 = 140 , \text{cm}^2 \]
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Area of the two triangular faces with base 12 cm: For these triangles, the height from the apex to the base of 12 cm is 10.6 cm. The area \( A_{triangle2} \) of one triangle is: \[ A_{triangle2} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 12 , \text{cm} \times 10.6 , \text{cm} = \frac{1}{2} \times 12 \times 10.6 = 63.6 , \text{cm}^2 \] Since there are two such triangles, their total area will be: \[ A_{triangles2} = 2 \times A_{triangle2} = 2 \times 63.6 , \text{cm}^2 = 127.2 , \text{cm}^2 \]
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Calculate the total surface area: To find the total surface area \( A_{total} \), we add the area of the base to the total area of the lateral faces: \[ A_{total} = A_{base} + A_{triangles1} + A_{triangles2} = 168 , \text{cm}^2 + 140 , \text{cm}^2 + 127.2 , \text{cm}^2 \] \[ A_{total} = 168 , \text{cm}^2 + 140 , \text{cm}^2 + 127.2 , \text{cm}^2 = 435.2 , \text{cm}^2 \]
Therefore, the surface area of the rectangular pyramid is 435.2 cm².