To find the surface area of a rectangular pyramid, you can use the formula:
\[ \text{Surface Area} = \text{Base Area} + \text{Lateral Area} \]
- Calculate the Base Area:
The base of the pyramid is a rectangle, and its area can be calculated using:
\[ \text{Base Area} = l \times w = 8 , \text{cm} \times 4 , \text{cm} = 32 , \text{cm}^2 \]
- Calculate the Slant Height:
To find the lateral area, we first need the slant height (\(s\)). The slant height can be calculated using the Pythagorean theorem. The height (\(h\)) is given as 2 cm, and since the base is 8 cm long, we need half of the base length:
\[ \text{Half base} = \frac{l}{2} = \frac{8}{2} = 4 , \text{cm} \]
Now we can find the slant height (\(s\)):
\[ s = \sqrt{h^2 + \left(\frac{l}{2}\right)^2} = \sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \approx 4.47 , \text{cm} \]
- Calculate the Lateral Area:
The lateral area consists of the areas of the four triangular faces. The area of each triangular face can be calculated using the formula:
\[ \text{Area of one triangular face} = \frac{1}{2} \times \text{base} \times \text{height} \]
For the two triangles with base 8 cm:
\[ \text{Area} = \frac{1}{2} \times 8 \times 4.47 \approx 17.88 , \text{cm}^2 \]
For the two triangles with base 4 cm:
\[ \text{Area} = \frac{1}{2} \times 4 \times 4.47 \approx 8.94 , \text{cm}^2 \]
So the total lateral area is:
\[ \text{Total Lateral Area} = 2 \times 17.88 + 2 \times 8.94 = 35.76 + 17.88 = 53.64 , \text{cm}^2 \]
- Calculate Total Surface Area:
Now add the base area to the lateral area:
\[ \text{Surface Area} = \text{Base Area} + \text{Lateral Area} = 32 + 53.64 = 85.64 , \text{cm}^2 \]
Thus, the surface area of the rectangular pyramid is:
\[ \boxed{85.64 , \text{cm}^2} \]
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