To find the length of one of the sides of the square pyramid, we need to understand a few components of the surface area and how they relate to one another.
The total surface area \( S \) of a square pyramid can be expressed as:
\[ S = \text{Base Area} + \text{Lateral Area} \]
In your case, you have:
- Total Surface Area \( S = 160 , \text{cm}^2 \)
- Lateral Area (area of the triangular faces) = \( 92 , \text{cm}^2 \)
First, we can find the area of the base:
\[ \text{Base Area} = S - \text{Lateral Area} = 160 , \text{cm}^2 - 92 , \text{cm}^2 = 68 , \text{cm}^2 \]
Since the base of the pyramid is a square, we can denote the length of one side of the square as \( s \). The area of the square base is given by:
\[ \text{Base Area} = s^2 \]
Setting this equal to the area we found:
\[ s^2 = 68 , \text{cm}^2 \]
To find \( s \), we take the square root of both sides:
\[ s = \sqrt{68} \approx 8.246 , \text{cm} \]
Therefore, the length of one of the sides of the square pyramid is approximately \( 8.25 , \text{cm} \).