To find the surface area of a rectangular pyramid, we can use the formula:
\[ \text{Surface Area} = \text{Base Area} + \text{Lateral Area} \]
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Calculate the base area:
The base of the pyramid is a rectangle, so the area of the base (A) can be calculated as: \[ A = l \times w \] Where \( l = 13 , \text{cm} \) and \( w = 11 , \text{cm} \): \[ A = 13 \times 11 = 143 , \text{cm}^2 \] -
Calculate the slant height:
To find the lateral area, we first need to determine the slant height (s) of the pyramid. The slant height can be found using the Pythagorean theorem. The height of the pyramid is given as \( h = 8 , \text{cm} \). The distance from the center of the base to the midpoint of one of the edges of the base can be calculated as: \[ \text{half width} = \frac{w}{2} = \frac{11}{2} = 5.5 , \text{cm} \]Now, applying the Pythagorean theorem: \[ s = \sqrt{h^2 + \left(\frac{w}{2}\right)^2} = \sqrt{8^2 + 5.5^2} = \sqrt{64 + 30.25} = \sqrt{94.25} \] Approximate \( s \): \[ s \approx 9.71 , \text{cm} \quad (\text{after calculating } \sqrt{94.25}) \]
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Calculate the lateral area:
The lateral area can be found by calculating the area of the four triangular faces. Each triangular face has a base that equals the width or length of the rectangle, and a height that equals the slant height.-
Two triangles have bases of length \( l = 13 , \text{cm} \): \[ \text{Area of two triangles with base } l = 2 \times \frac{1}{2} \times l \times s = 2 \times \frac{1}{2} \times 13 \times 9.71 = 126.23 , \text{cm}^2 \]
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Two triangles have bases of width \( w = 11 , \text{cm} \): \[ \text{Area of two triangles with base } w = 2 \times \frac{1}{2} \times w \times s = 2 \times \frac{1}{2} \times 11 \times 9.71 = 106.81 , \text{cm}^2 \]
Combine both triangular areas for the total lateral area: \[ \text{Lateral Area} = 126.23 + 106.81 = 233.04 , \text{cm}^2 \]
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Calculate the total surface area:
\[ \text{Surface Area} = \text{Base Area} + \text{Lateral Area} = 143 + 233.04 = 376.04 , \text{cm}^2 \]
Thus, the surface area of the rectangular pyramid is approximately: \[ \boxed{376.04 , \text{cm}^2} \]
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