A delicious circular pie with diameter 12 cm is cut into three equal-sized sector-shaped pieces. Let l be the number of centimeters in the length of the longest line segment that may be drawn in one of these pieces. What is l^2?

The longest line segment that can be drawn inside an arc is the red line below:

Diameter = 12cm --> Radius = 6cm

Central angle of one arc = 360 deg / 3 = 120 deg

l^2 = 6^2 + 6^2 - 2(6)(6)cos120 deg (Law of Cosines)

I^2 = 72(1-cos(120)) = 108.