To calculate the surface area of a square pyramid, we need to add the area of the base and the area of the four triangular faces.
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Calculate the area of the base:
Since the base is a square, the area is given by the formula: \[ \text{Area of the base} = \text{side length}^2 \] You mentioned the area of the base of a rectangular prism is \(20 \times 18\), which suggests the dimensions of the base of the pyramid could also be derived from these dimensions. However, we would need confirmation if the base of the pyramid is, in fact, square.For a square base (let's assume the side length is "a"): \[ \text{Area of the base} = a^2 \]
If the base is rectangular, we would need the dimensions to find the area accurately.
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Calculate the area of the triangular faces:
Each triangular face can be calculated using the formula for the area of a triangle: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] You provided the base length and slant height for two triangles.For the first triangle (base = 20, slant height = 16.6): \[ \text{Area}_1 = \frac{1}{2} \times 20 \times 16.6 = 166 \]
For the second triangle (base = 18, slant height = 17.2): \[ \text{Area}_2 = \frac{1}{2} \times 18 \times 17.2 = 154.8 \]
Assuming there are two more triangular faces that are congruent to these two (which would be typical if the base is rectangular and not a square), then: \[ \text{Total triangular area} = 2 \times \text{Area}_1 + 2 \times \text{Area}_2 = 2 \times 166 + 2 \times 154.8 = 332 + 309.6 = 641.6 \]
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Adding the areas for total surface area:
- If the area of the base is derived from the rectangle as suggested (\(20 \times 18 = 360\)), then: \[ \text{Surface Area} = \text{Area of the base} + \text{Total triangular area} = 360 + 641.6 = 1001.6 \]
However, if we align with the most common calculation for a square pyramid and you provided heights and bases that seem more to match a singular square case (for variations with same base), please clarify the base dimensions.
To summarize, if base is square or rectangular and specifying dimensions given:
- Please clarify whether the base was indeed a square and what dimensions to use if not.
- Without confirmation on the dimensions of the base, the response cannot match the given options (522 ft², 148.5 ft², etc.).
Please provide clarification on how you wish to pursue the dimension analysis.