To calculate the surface area of the rectangular pyramid, we need to find the area of the base and the areas of the triangular faces.
Step 1: Calculate the area of the base
The base of the pyramid is a rectangle with the length (L) and width (W):
\[ \text{Area of the base} = L \times W = 12 , \text{cm} \times 14 , \text{cm} = 168 , \text{cm}^2 \]
Step 2: Calculate the area of the triangular faces
The pyramid has 4 triangular faces. We'll consider two pairs of triangular faces, one pair for each pair of sides (length and width).
Area of the triangular face on the width side:
- Base = Width (W = 14 cm)
- Height = Height from the apex of the pyramid to the midpoint of the width side (H = 10 cm)
Using the formula for the area of a triangle:
\[ \text{Area of one triangular face (width side)} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 14 , \text{cm} \times 10 , \text{cm} = 70 , \text{cm}^2 \]
Since there are two triangular faces on the width side:
\[ \text{Total area (width side)} = 2 \times 70 , \text{cm}^2 = 140 , \text{cm}^2 \]
Area of the triangular face on the length side:
- Base = Length (L = 12 cm)
- Height = Height from the apex of the pyramid to the midpoint of the length side (H = 10.6 cm)
Using the same triangle area formula:
\[ \text{Area of one triangular face (length side)} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times 12 , \text{cm} \times 10.6 , \text{cm} = 63.6 , \text{cm}^2 \]
Since there are two triangular faces on the length side:
\[ \text{Total area (length side)} = 2 \times 63.6 , \text{cm}^2 = 127.2 , \text{cm}^2 \]
Step 3: Calculate the total surface area
Now, add the area of the base and the areas of the triangular faces:
\[ \text{Total Surface Area} = \text{Area of base} + \text{Total area (width side)} + \text{Total area (length side)} \] \[ \text{Total Surface Area} = 168 , \text{cm}^2 + 140 , \text{cm}^2 + 127.2 , \text{cm}^2 = 435.2 , \text{cm}^2 \]
Conclusion
The surface area of the rectangular pyramid is \( \boxed{435.2 , \text{cm}^2} \).