To find the surface area of a rectangular pyramid, you need to calculate the areas of its base and its triangular faces.
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Base Area: The base is a rectangle with length \( l \) and width \( w \). \[ \text{Base Area} = l \times w = 13 , \text{cm} \times 11 , \text{cm} = 143 , \text{cm}^2 \]
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Triangular Faces: The pyramid has four triangular faces. To find the area of the triangular faces, we need to calculate the slant heights of the triangular sides.
- First, calculate the slant heights of the triangular faces:
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The height of one triangular face from the center of the base to the apex (the top of the pyramid) can be found using the Pythagorean theorem. The height of the pyramid \( h = 8 , \text{cm} \), and the horizontal distance from the center of the base to the midpoint of a side is half of the base’s width or length.
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For the triangles with base \( w \):
- Horizontal distance to the midpoint of the width \( = \frac{w}{2} = \frac{11}{2} = 5.5 , \text{cm} \)
- The slant height \( s_1 \) from the apex to this midpoint is: \[ s_1 = \sqrt{h^2 + \left(\frac{w}{2}\right)^2} = \sqrt{8^2 + 5.5^2} = \sqrt{64 + 30.25} = \sqrt{94.25} \approx 9.71 , \text{cm} \]
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For the triangles with base \( l \):
- Horizontal distance to the midpoint of the length \( = \frac{l}{2} = \frac{13}{2} = 6.5 , \text{cm} \)
- The slant height \( s_2 \) from the apex to this midpoint is: \[ s_2 = \sqrt{h^2 + \left(\frac{l}{2}\right)^2} = \sqrt{8^2 + 6.5^2} = \sqrt{64 + 42.25} = \sqrt{106.25} \approx 10.31 , \text{cm} \]
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- First, calculate the slant heights of the triangular faces:
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Area of Each Triangular Face:
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There are 2 triangles with base \( w \) and height \( s_1 \): \[ \text{Area of one triangle with base } w = \frac{1}{2} \times w \times s_1 = \frac{1}{2} \times 11 \times 9.71 \approx 53.405 , \text{cm}^2 \] \[ \text{Total area for two triangles with base } w = 2 \times 53.405 \approx 106.81 , \text{cm}^2 \]
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There are 2 triangles with base \( l \) and height \( s_2 \): \[ \text{Area of one triangle with base } l = \frac{1}{2} \times l \times s_2 = \frac{1}{2} \times 13 \times 10.31 \approx 67.015 , \text{cm}^2 \] \[ \text{Total area for two triangles with base } l = 2 \times 67.015 \approx 134.03 , \text{cm}^2 \]
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Total Surface Area: \[ \text{Total Surface Area} = \text{Base Area} + \text{Total Area for triangles with base } w + \text{Total Area for triangles with base } l \] \[ \text{Total Surface Area} = 143 + 106.81 + 134.03 = 383.84 , \text{cm}^2 \]
Therefore, the surface area of the rectangular pyramid is approximately \( \boxed{383.84} , \text{cm}^2 \).
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