To find the length of one side of the square pyramid based on the given surface area and the area of the lateral triangles, we can follow these steps:
-
Understanding the Surface Area: The total surface area \( S \) of a square pyramid can be expressed as the sum of the area of the base and the area of the lateral faces (triangular sides).
\[ S = \text{Area of base} + \text{Area of lateral triangles} \]
-
Given Values:
- Total surface area \( S = 160 , \text{cm}^2 \)
- Area of lateral triangles = \( 96 , \text{cm}^2 \)
-
Finding the Area of the Base: We can find the area of the base by rearranging the equation:
\[ \text{Area of base} = S - \text{Area of lateral triangles} \] \[ \text{Area of base} = 160 , \text{cm}^2 - 96 , \text{cm}^2 = 64 , \text{cm}^2 \]
-
Area of Base of a Square Pyramid: The area of the base of a square pyramid is given by:
\[ \text{Area of base} = s^2 \]
where \( s \) is the length of one side of the base (which is a square). We now have:
\[ s^2 = 64 , \text{cm}^2 \]
-
Finding the Length of One Side: Taking the square root of both sides gives:
\[ s = \sqrt{64} , \text{cm} = 8 , \text{cm} \]
Therefore, the length of one side of the base of the square pyramid is 8 cm.