To find the surface area of a rectangular pyramid, we need to calculate the area of the base and the area of the four triangular faces.
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Area of the base (rectangular): \[ \text{Area of the base} = l \times w = 15 , \text{m} \times 12 , \text{m} = 180 , \text{m}^2 \]
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Area of the triangular faces: To calculate the area of the triangular faces, we need to find the slant heights of the triangles.
First, we can calculate the lengths from the center of the base to the vertices: \[ \text{Half of length} = \frac{l}{2} = \frac{15}{2} = 7.5 , \text{m} \] \[ \text{Half of width} = \frac{w}{2} = \frac{12}{2} = 6 , \text{m} \]
Next, we find the slant height \( s \) using the Pythagorean theorem: \[ s = \sqrt{\left(\frac{l}{2}\right)^2 + h^2} = \sqrt{(7.5)^2 + (10)^2} = \sqrt{56.25 + 100} = \sqrt{156.25} = 12.5 , \text{m} \]
The triangular sides are isosceles triangles. The area of one triangular face is: \[ \text{Area of one triangular face} = \frac{1}{2} \times \text{base} \times \text{height} \] Here, the base can be either length or width depending on which triangular face you are calculating. Remember, each triangular face base is either 15 m or 12 m.
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For the two triangular faces with base = 15 m: \[ \text{Area} = \frac{1}{2} \times 15 \times 12.5 = \frac{1}{2} \times 15 \times 12.5 = 93.75 , \text{m}^2 \]
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For the two triangular faces with base = 12 m: \[ \text{Area} = \frac{1}{2} \times 12 \times 12.5 = \frac{1}{2} \times 12 \times 12.5 = 75 , \text{m}^2 \]
Since there are two of each triangular face: \[ \text{Total area of triangular faces} = 2 \times 93.75 + 2 \times 75 = 187.5 + 150 = 337.5 , \text{m}^2 \]
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Total surface area: Finally, we add the area of the base to the total area of the triangular faces: \[ \text{Total Surface Area} = \text{Area of base} + \text{Total area of triangular faces} \] \[ = 180 + 337.5 = 517.5 , \text{m}^2 \]
Thus, the surface area of the rectangular pyramid is 517.50 m².
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