Notes
four hexagons solution

Solution to the Four Hexagons Puzzle

Solution by Transformations

In the above diagram, rhombus $A B C D$ transforms to $A E F B$ through a rotation of $60^\circ$ clockwise. This same transformation takes the point $G$ to point $H$. So $G C$ is transformed to $H F$ and hence these have the same length. The length of $G H$ is twice the length of a side of the smallest hexagon, so the area of the pink hexagon is four times the area of the smallest, hence is $48$.