The square and the circle share an identical diameter, denoted as \(d\).
Area of the square land: The side length of the square is equal to its diameter \(d\). The area of the square is \(d^2\).
Area of the circular land: The area of a circle is calculated as \(\pi r^2\), where \(r\) is the radius. Given that the radius \(r = \frac{d}{2}\), the area of the circle is \(\pi \left(\frac{d}{2}\right)^2 = \frac{\pi d^2}{4}\).
Ratio of Areas:
\[ \text{Ratio} = \frac{\text{Area of square}}{\text{Area of circle}} = \frac{d^2}{\frac{\pi d^2}{4}} = \frac{4}{\pi}. \]
Consequently, the ratio of their areas is \(4 : \pi\).