To show the ratio of the area of the inscribed square to the area of the circle, we first determine the dimensions of the square. When a square is inscribed in a circle of radius $r$, the diagonal of the square passes through the centre of the circle. Therefore, the diagonal of the square is equal to the diameter of the circle, which is $2r$.

Let the side of the square be $a$. Using the Pythagorean theorem, the relationship between the side and the diagonal of a square is:

$$a^2 + a^2 = (\text{diagonal})^2$$

Substituting the diagonal as $2r$, we get:

$$2a^2 = (2r)^2 = 4r^2$$

Solving for $a^2$, we find:

$$a^2 = 2r^2$$

Since the area of the square is $a^2$, it equals $2r^2$.

From the provided context, we know the area of the circle is $\pi r^2$.

Now, we find the ratio of the area of the square to the area of the circle:

$$\text{Ratio} = \frac{\text{Area of square}}{\text{Area of circle}} = \frac{2r^2}{\pi r^2} = \frac{2}{\pi}$$

Finally, calculating the decimal value of this ratio, we get:

$$\frac{2}{\pi} \approx \frac{2}{3.14159} \approx 0.637$$

Thus, the ratio of the area of the square to the area of the circle is equal to $\frac{2}{\pi} \approx 0.637$.