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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First, calculate the area of the base. Since the base is a rectangle: \[ \text{Base Area} = l \times w = 8 , \text{cm} \times 4 , \text{cm} = 32 , \text{cm}^2 \]
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Next, we need to find the lateral area. A rectangular pyramid has four triangular faces. The lateral area can be calculated by finding the area of each triangular face.
The height of each triangular face is measured from the apex (top) of the pyramid to the midpoint of the base edge. We need to calculate the slant heights for the triangular faces.
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For the triangular faces with a base of length \(l = 8\) cm, we find the slant height \(s_1\) from the apex to the midpoint of the base edge.
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The distance from the apex vertically downward to the base is \(h = 2\) cm.
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The distance from the midpoint of the base edge (for the length 8 cm) to the center line (in the width direction) is half of width \(w = 4\) cm, or \(2\) cm.
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Using the Pythagorean theorem: \[ s_1 = \sqrt{h^2 + \left(\frac{l}{2}\right)^2} = \sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} \approx 4.47 , \text{cm} \]
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For the triangular faces with a base of width \(w = 4\) cm (similar steps apply): \[ s_2 = \sqrt{h^2 + \left(\frac{w}{2}\right)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} \approx 2.83 , \text{cm} \]
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Now we can find the areas of the triangular faces.
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The area of the two triangular faces with base \(l = 8\) cm and height \(s_1 \approx 4.47\) cm: \[ \text{Area}_{l} = 2 \times \left(\frac{1}{2} \times 8 \times 4.47\right) = 2 \times 17.88 \approx 35.76 , \text{cm}^2 \]
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The area of the two triangular faces with base \(w = 4\) cm and height \(s_2 \approx 2.83\) cm: \[ \text{Area}_{w} = 2 \times \left(\frac{1}{2} \times 4 \times 2.83\right) = 2 \times 5.66 \approx 11.32 , \text{cm}^2 \]
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Combine the areas to find the total lateral area: \[ \text{Lateral Area} = \text{Area}{l} + \text{Area}{w} = 35.76 , \text{cm}^2 + 11.32 , \text{cm}^2 \approx 47.08 , \text{cm}^2 \]
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Finally, add the base area to the lateral area to find the total surface area: \[ \text{Surface Area} = \text{Base Area} + \text{Lateral Area} = 32 , \text{cm}^2 + 47.08 , \text{cm}^2 \approx 79.08 , \text{cm}^2 \]
So, the surface area of the pyramid is approximately 79.08 cm².
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