To find the length of one of the sides of the square pyramid, we have the following information:
- The total surface area of the square pyramid is \( 160 , \text{cm}^2 \).
- The area of the lateral triangles (the triangular faces) is \( 96 , \text{cm}^2 \).
The total surface area of a square pyramid can be calculated as:
\[ \text{Total Surface Area} = \text{Base Area} + \text{Lateral Area} \]
In this case, the base of the pyramid is a square. Let \( s \) be the length of one side of the base.
The area of the base (which is a square) is:
\[ \text{Base Area} = s^2 \]
The total lateral area is given as \( 96 , \text{cm}^2 \). Therefore, we can write:
\[ \text{Total Surface Area} = s^2 + \text{Lateral Area} \]
Substituting the known values into the equation yields:
\[ 160 = s^2 + 96 \]
Now, we can solve for \( s^2 \):
\[ s^2 = 160 - 96 \] \[ s^2 = 64 \]
Now, taking the square root of both sides gives:
\[ s = \sqrt{64} \] \[ s = 8 , \text{cm} \]
Thus, the length of one of the sides of the base of the square pyramid is \( \boxed{8} , \text{cm} \).