To find the surface area of a rectangular pyramid, you need to calculate the area of the base and the area of the four triangular faces.
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Calculate the area of the base (which is a rectangle): \[ \text{Area of the base} = \text{length} \times \text{width} = l \times w = 13 , \text{cm} \times 11 , \text{cm} = 143 , \text{cm}^2 \]
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Calculate the slant height (lateral height) of the triangular faces: The slant height can be found using the Pythagorean theorem. You can find the slant heights relative to each direction (length and width).
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For the triangular face using length (l): \[ \text{slant height } h_l = \sqrt{(h^2) + \left(\frac{w}{2}\right)^2} = \sqrt{(8^2) + \left(\frac{11}{2}\right)^2} = \sqrt{64 + 30.25} = \sqrt{94.25} \approx 9.71 , \text{cm} \]
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For the triangular face using width (w): \[ \text{slant height } h_w = \sqrt{(h^2) + \left(\frac{l}{2}\right)^2} = \sqrt{(8^2) + \left(\frac{13}{2}\right)^2} = \sqrt{64 + 42.25} = \sqrt{106.25} \approx 10.31 , \text{cm} \]
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Calculate the area of each of the triangular faces:
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There are two triangular faces on the length sides: \[ \text{Area of one triangular face (length)} = \frac{1}{2} \times w \times h_l = \frac{1}{2} \times 11 \times 9.71 \approx 53.41 , \text{cm}^2 \] Since there are two of these triangles: \[ \text{Total Area for length sides} = 2 \times 53.41 \approx 106.82 , \text{cm}^2 \]
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There are also two triangular faces on the width sides: \[ \text{Area of one triangular face (width)} = \frac{1}{2} \times l \times h_w = \frac{1}{2} \times 13 \times 10.31 \approx 67.01 , \text{cm}^2 \] Since there are two of these triangles: \[ \text{Total Area for width sides} = 2 \times 67.01 \approx 134.02 , \text{cm}^2 \]
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Combine all areas to find the total surface area: \[ \text{Total Surface Area} = \text{Area of the base} + \text{Total Area for length sides} + \text{Total Area for width sides} \] \[ \text{Total Surface Area} = 143 + 106.82 + 134.02 \approx 383.84 , \text{cm}^2 \]
Thus, the surface area of the rectangular pyramid is approximately 383.84 cm².