To calculate the surface area of a rectangular pyramid, we need to find the area of the base and the areas of the four triangular faces.
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Calculate the area of the base: The base is a rectangle with length \( l = 8 \) cm and width \( w = 4 \) cm. \[ \text{Area of the base} = l \times w = 8 , \text{cm} \times 4 , \text{cm} = 32 , \text{cm}^2 \]
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Calculate the area of the triangular faces: The pyramid has two pairs of triangular faces.
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For the triangular face on the length side (base = 8 cm):
- The slant height can be calculated using the Pythagorean theorem. The height of the pyramid is \( h = 2 \) cm, and half the base length (for the triangle) is \( \frac{l}{2} = \frac{8}{2} = 4 \) cm. \[ \text{Slant height for this triangle} = \sqrt{\left(\frac{l}{2}\right)^2 + h^2} = \sqrt{4^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \approx 4.472 , \text{cm} \]
- Area of one triangular face: \[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \times 4.472 \approx 17.888 , \text{cm}^2 \]
- Since there are two such faces: \[ \text{Total area from length triangles} = 2 \times 17.888 \approx 35.776 , \text{cm}^2 \]
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For the triangular face on the width side (base = 4 cm):
- Similarly, the slant height is: \[ \text{Slant height for this triangle} = \sqrt{\left(\frac{w}{2}\right)^2 + h^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \approx 2.828 , \text{cm} \]
- Area of one triangular face: \[ \text{Area} = \frac{1}{2} \times 4 \times 2.828 \approx 5.656 , \text{cm}^2 \]
- Since there are two such faces: \[ \text{Total area from width triangles} = 2 \times 5.656 \approx 11.312 , \text{cm}^2 \]
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Calculate total surface area: \[ \text{Total Surface Area} = \text{Area of the base} + \text{Total area from length triangles} + \text{Total area from width triangles} \] \[ \text{Total Surface Area} = 32 + 35.776 + 11.312 \approx 79.088 , \text{cm}^2 \]
Rounding to the nearest hundredth: \[ \text{Total Surface Area} \approx 79.09 , \text{cm}^2 \]
Thus, the surface area of the rectangular pyramid is approximately 79.09 cm².