To find the length of one of the sides of the square pyramid, we can use the information given about the surface area and the lateral area.
- Total Surface Area of the Pyramid (A): This is given as 160 cm².
- Lateral Area (LA): The area of the lateral triangles is given as 96 cm².
The total surface area of a square pyramid consists of the lateral area plus the area of the base (which is a square):
\[ A = LA + \text{Area of Base} \]
We can rearrange this to find the area of the base:
\[ \text{Area of Base} = A - LA \]
Substituting the known values:
\[ \text{Area of Base} = 160 , \text{cm}^2 - 96 , \text{cm}^2 = 64 , \text{cm}^2 \]
Since the base of the pyramid is a square, we can express the area of the base as:
\[ \text{Area of Base} = s^2 \]
where \(s\) is the length of one side of the square base. Therefore, we can set up the equation:
\[ s^2 = 64 , \text{cm}^2 \]
Taking the square root of both sides, we find:
\[ s = \sqrt{64} = 8 , \text{cm} \]
Thus, the length of one side of the square base is 8 cm.