To calculate the surface area of a rectangular pyramid, you need to consider both the area of the base and the area of the triangular faces.
Step 1: Calculate the Area of the Base
The base of the pyramid is rectangular with a length of \( l = 6 \) cm. For simplicity, let's assume it's also 6 cm wide (i.e., it is a square base, but you can adjust if provided with a different width). Thus, the area of the base \( A_{base} \) is:
\[ A_{base} = l \times w = 6 , \text{cm} \times 6 , \text{cm} = 36 , \text{cm}^2 \]
Step 2: Calculate the Area of the Triangular Faces
The pyramid has four triangular faces. The area of each triangular face can be calculated using the formula:
\[ A_{triangle} = \frac{1}{2} \times \text{base} \times \text{height} \]
The height of the triangles can be found using the Pythagorean theorem. The "base" of each triangular face is one side of the base of the pyramid, and the height is the slant height of the triangle. Since the pyramid is symmetrical, if we take one side of the base, the height can be calculated as follows:
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Find the slant height \( s \). For a triangular face with a height of 8 cm and half of the base, the length will be \( \frac{6}{2} = 3 \) cm.
Using the Pythagorean theorem:
\[ s = \sqrt{\left(\frac{6}{2}\right)^2 + 8^2} = \sqrt{3^2 + 8^2} = \sqrt{9 + 64} = \sqrt{73} , \text{cm} \]
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Calculate the area of one triangular face:
Now, use this slant height for the calculation. Each of the triangular faces will have this height:
\[ A_{triangle} = \frac{1}{2} \times 6 , \text{cm} \times \sqrt{73} , \text{cm} \]
Thus, the area of one triangular face is:
\[ A_{triangle} = 3 \sqrt{73} , \text{cm}^2 \]
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Calculate the area of all four triangular faces:
\[ A_{triangles} = 4 \times A_{triangle} = 4 \times 3 \sqrt{73} , \text{cm}^2 = 12 \sqrt{73} , \text{cm}^2 \]
Step 3: Total Surface Area
Finally, add the area of the base and the area of the triangular faces:
\[ \text{Surface Area} = A_{base} + A_{triangles} = 36 + 12\sqrt{73} , \text{cm}^2 \]
Numerical Approximation
Calculating \( \sqrt{73} \) is approximately 8.54:
\[ 12 \times \sqrt{73} \approx 12 \times 8.54 \approx 102.48 \]
So the total surface area will be approximately:
\[ \text{Surface Area} \approx 36 + 102.48 \approx 138.48 , \text{cm}^2 \]
Therefore, the surface area of the rectangular pyramid is approximately 138.48 cm².