To find the ratio of the perimeters of the smaller square to the larger square, we first need to calculate the perimeter of each square.
The formula for the perimeter \( P \) of a square with side length \( s \) is given by:
\[ P = 4s \]
For the smaller square with a side length of 2 centimeters:
\[ P_{\text{smaller}} = 4 \times 2 = 8 \text{ centimeters} \]
For the larger square with a side length of 6 centimeters:
\[ P_{\text{larger}} = 4 \times 6 = 24 \text{ centimeters} \]
Now, we can find the ratio of the perimeter of the smaller square to the perimeter of the larger square:
\[ \text{Ratio} = \frac{P_{\text{smaller}}}{P_{\text{larger}}} = \frac{8}{24} \]
To reduce the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 8:
\[ \frac{8 \div 8}{24 \div 8} = \frac{1}{3} \]
Thus, the ratio of the perimeter of the smaller square to the larger square is:
\[ \boxed{\frac{1}{3}} \]
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