To find the surface area of a rectangular pyramid, we need to calculate the areas of the base and the four triangular faces.
Given dimensions:
- Base length \( L = 12 , \text{cm} \)
- Base width \( W = 14 , \text{cm} \)
- Height on the side of the width \( H_1 = 10 , \text{cm} \)
- Height on the side of the length \( H_2 = 10.6 , \text{cm} \)
Step 1: Calculate the area of the base
The area of the rectangular base \( A_{\text{base}} \) is given by: \[ A_{\text{base}} = L \times W = 12 , \text{cm} \times 14 , \text{cm} = 168 , \text{cm}^2 \]
Step 2: Calculate the areas of the triangular faces
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Triangular face on the side of width (height \(H_1\)): The area of a triangle is given by: \[ A_{\text{triangle-1}} = \frac{1}{2} \times \text{Base} \times \text{Height} \] Using the width as the base: \[ A_{\text{triangle-1}} = \frac{1}{2} \times W \times H_1 = \frac{1}{2} \times 14 , \text{cm} \times 10 , \text{cm} = 70 , \text{cm}^2 \]
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Triangular face on the side of length (height \(H_2\)): Using the length as the base: \[ A_{\text{triangle-2}} = \frac{1}{2} \times L \times H_2 = \frac{1}{2} \times 12 , \text{cm} \times 10.6 , \text{cm} = 63.6 , \text{cm}^2 \]
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There are two identical triangular faces for each side: Thus, we need to account for two of each triangular face: \[ \text{Total triangular area} = 2 \times A_{\text{triangle-1}} + 2 \times A_{\text{triangle-2}} = 2 \times 70 , \text{cm}^2 + 2 \times 63.6 , \text{cm}^2 \] \[ = 140 , \text{cm}^2 + 127.2 , \text{cm}^2 = 267.2 , \text{cm}^2 \]
Step 3: Calculate the total surface area
The total surface area \( A_{\text{total}} \) is the sum of the area of the base and the area of the triangular faces: \[ A_{\text{total}} = A_{\text{base}} + \text{Total triangular area} = 168 , \text{cm}^2 + 267.2 , \text{cm}^2 = 435.2 , \text{cm}^2 \]
Final Answer
Thus, the surface area of the rectangular pyramid is: \[ \boxed{435.2} , \text{cm}^2 \]