4. As shown in the figure, there is a circle and three squares. The vertices of the middle square are all on the circle. The intersection points of the circle with the largest square and the vertices of the smallest square are the midpoints of their respective line segments. The area of the largest square is 12 square centimeters. What is the area of the smallest square in square centimeters?
4. As shown in the figure, there is a circle and three squares. The vertices of the middle square are all on the circle. The intersection points of the circle with the largest square and the vertices of the smallest square are the midpoints of their respective line segments. The area of the largest square is 12 square centimeters. What is the area of the smallest square in square centimeters?

Rotate the original figure around the center $O$ on the circular surface so that point $P$ coincides with $E$, point $Q$ coincides with $F$, point $S$ coincides with $G$, and point $T$ coincides with $H$, forming the figure above. Since $P Q T S$ are the midpoints of their respective line segments, it is easy to see that $S_{P Q S T}=\frac{1}{2} S_{A B C D}$. Similarly, we can get $S_{I J K L}=\frac{1}{2} S_{P Q S T}$, and then further solve it.

【Solution】According to the analysis, we have:
$$
\begin{array}{l}
S_{I J K L}=\frac{1}{2} S_{P Q S T} \\
=\frac{1}{2}\left(\frac{1}{2} S_{A B C D}\right) \\
=\frac{1}{4} S_{A B C D} \\
=\frac{1}{4} \times 12 \\
=3 \text { (square centimeters) }
\end{array}
$$

Answer: The area of the smallest square is 3 square centimeters.