Which Area is the Largest??

Make $6$ cuts through the center of the figure as follows:

Now there are $12$ blue triangles, $12$ orange triangles, and $12$ green triangles that are all congruent, so the areas are equal.

Let the are of a blue triangle be $2\alpha$ . If we cut the green triangle into half and fold it along the line it shares with the blue triangle, it fits nicely as half of the blue triangle. Therefore the area of the green triangle is the same as that of the blue triangle $2\alpha$ . Since there are both six blue and green triangles the blue area and green area are equal.

We also note that the six blue triangles when fold along the line they share with the orange hexagon, fit nicely to form the orange hexagon. Therefore the orange area is equal to the blue area.

Therefore, all areas are equal .