jmc

First of all, let us draw one of these pieces in question, labeling points of interest as necessary: We can see that the longest segment we can draw is from $A$ to $B,$ and to find $AB,$ we should create right triangles by drawing the perpendicular bisector to $AB.$ Since $\angle MOB$ is half of $\angle AOB,$ which is a third of a full circle, we have $\angle MOB=60^{\circ },$ so $△MOB$ is a 30-60-90 triangle. Since the diameter of the pie is $12\text{ cm},$ we see that $OB=6\text{ cm},$ so $MO=3\text{ cm}$ and $MB=3\sqrt{3}\text{ cm}.$ Then, $AB=2\cdot MB=6\sqrt{3}\text{ cm},$ so $l=6\sqrt{3}.$ Finally, $l^2=\boxed{108}.$