To find the surface area of a rectangular pyramid, we first need to understand the formula for the surface area. The surface area \( SA \) can be calculated with the following steps:
- Calculate the area of the base.
- Calculate the area of the triangular faces.
- Add both areas together.
Step 1: Area of the Base
The base of the pyramid is a rectangle. The area \( A_b \) of the base can be calculated using the formula:
\[ A_b = l \times w \]
Substituting the given dimensions \( l = 8 , \text{cm} \) and \( w = 4 , \text{cm} \):
\[ A_b = 8 \times 4 = 32 , \text{cm}^2 \]
Step 2: Area of the Triangular Faces
A rectangular pyramid has 4 triangular faces. To find the area of each triangular face, we need to calculate the slant heights for each face.
Finding the Slant Heights
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For the triangular faces with the base of length \( l \):
The slant height \( s_l \) for these is the distance from the apex of the pyramid (the vertex opposite the base) to the midpoint of the base edge. The midpoint of the base edge will be at \( \frac{l}{2} = \frac{8}{2} = 4 , \text{cm} \) away from the edge along the length direction.
To find the slant height, we can use the Pythagorean theorem:
\[ s_l = \sqrt{h^2 + \left(\frac{l}{2}\right)^2} = \sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \approx 4.47 , \text{cm} \]
The area \( A_{tr1} \) of each of these triangles is:
\[ A_{tr1} = \frac{1}{2} \times l \times s_l = \frac{1}{2} \times 8 \times 4.47 \approx 17.88 , \text{cm}^2 \]
Since there are two such triangular faces, the total area from these faces is:
\[ A_{tr1 \text{ total}} = 2 \times 17.88 \approx 35.76 , \text{cm}^2 \]
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For the triangular faces with the base of width \( w \):
Similarly, the slant height \( s_w \) for these is:
\[ s_w = \sqrt{h^2 + \left(\frac{w}{2}\right)^2} = \sqrt{2^2 + 2^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \approx 2.83 , \text{cm} \]
The area \( A_{tr2} \) of each of these triangles is:
\[ A_{tr2} = \frac{1}{2} \times w \times s_w = \frac{1}{2} \times 4 \times 2.83 \approx 5.66 , \text{cm}^2 \]
Since there are two such triangular faces, the total area from these faces is:
\[ A_{tr2 \text{ total}} = 2 \times 5.66 \approx 11.32 , \text{cm}^2 \]
Step 3: Total Surface Area
Now we can add the base area and the area of the triangular faces:
\[ SA = A_b + A_{tr1 \text{ total}} + A_{tr2 \text{ total}} = 32 + 35.76 + 11.32 \approx 79.08 , \text{cm}^2 \]
Thus, the total surface area of the rectangular pyramid is approximately:
\[ \boxed{79.08} , \text{cm}^2 \]