jmc

The two right triangles on the ends of the rectangle can be pushed together to form an equilateral triangle that is identical to each of the other equilateral triangles in the diagram. So, $AB$ is equal to the total length of 3 side lengths of an equilateral triangle. Therefore, each side of each equilateral triangle has length $12/3=4$. Therefore, our problem is to find the total area of two equilateral triangles with side length 4.

Drawing an altitude of an equilateral triangle splits it into two 30-60-90 right triangles:

An altitude of an equilateral triangle is therefore $\sqrt{3}$ times the length of half the side length of the triangle. Therefore, an equilateral triangle with side length 4 has altitude length $\sqrt{3}(4/2)=2\sqrt{3}$, and area $(2\sqrt{3})(4)/2=4\sqrt{3}$ square units. The shaded regions consist of two of these equilateral triangles, so their total area is $2(4\sqrt{3})=\boxed{8\sqrt{3}}$.