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\text{Ratio} = \frac{A_{\text{circle}}}{A_{\text{octagon}}} = \frac{\pi x^2}{2(1 + \sqrt{2})x^2} = \frac{\pi}{2(1 + \sqrt{2})}
Thus, the ratio of the area of the circle to the area of the octagon is \(\boxed{\frac{\pi}{2(1 + \sqrt{2})}}\).
A computer scientist is designing an AI interface that uses equilateral triangular logic gates. Each gate has a side length of \( s \). If the side length is increased by 50%, what is the ratio of the new area to the original area of one triangular gate?

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