To find the surface area of a rectangular pyramid, we need to consider both the area of the base and the areas of the triangular faces.
Given:
- Base dimensions: 10 ft (longer side) and 7 ft (shorter side)
- Slant heights: 7 ft (for the side with length 10 ft) and 8 ft (for the side with length 7 ft)
Step 1: Calculate the area of the base.
The area \( A_{\text{base}} \) of the rectangular base is given by:
\[ A_{\text{base}} = \text{length} \times \text{width} = 10 , \text{ft} \times 7 , \text{ft} = 70 , \text{ft}^2 \]
Step 2: Calculate the area of the triangular faces.
There are four triangular faces corresponding to the sides of the base:
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Triangular Face corresponding to the 10 ft side (Height = 7 ft):
- Base of triangle = 10 ft
- Height of triangle = 7 ft
The area \( A_1 \) of this triangle is given by:
\[ A_1 = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10 , \text{ft} \times 7 , \text{ft} = 35 , \text{ft}^2 \]
-
Triangular Face corresponding to the 7 ft side (Height = 8 ft):
- Base of triangle = 7 ft
- Height of triangle = 8 ft
The area \( A_2 \) of this triangle is given by:
\[ A_2 = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 7 , \text{ft} \times 8 , \text{ft} = 28 , \text{ft}^2 \]
Since there are two triangular faces corresponding to each dimension:
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Two triangular faces with base = 10 ft: \[ A_{10} = 2 \times A_1 = 2 \times 35 , \text{ft}^2 = 70 , \text{ft}^2 \]
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Two triangular faces with base = 7 ft: \[ A_{7} = 2 \times A_2 = 2 \times 28 , \text{ft}^2 = 56 , \text{ft}^2 \]
Step 3: Calculate total surface area.
Now, we sum the area of the base and the areas of the triangular faces:
\[ \text{Total Surface Area} = A_{\text{base}} + A_{10} + A_{7} \]
\[ \text{Total Surface Area} = 70 , \text{ft}^2 + 70 , \text{ft}^2 + 56 , \text{ft}^2 = 196 , \text{ft}^2 \]
Thus, the total surface area of the rectangular pyramid is \( 196 , \text{ft}^2 \).
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