The diameters of two circles are the side of a square and the diagonal of the square. The ratio of the areas of the smaller circle and the larger circle is
Let's say side of square is $$x$$.
Hence length of diagonal will be $$\sqrt2x$$
Since these lengths  are diameters of circles, Hence ratio of their area would be
$$\pi(r_1)^2:\pi(r_2)^2$$(where$$r_1$$ is radius of smaller circle and $$r_2$$ is radius of larger circle)
or $$\frac{x^2}{4} :\frac{2x^2}{4}$$
or 1:2
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