A hexagon is six equilateral triangles; the apothem is the height of one of them.
The right triangle associated with that apothem has sides in ratio [tex]1 : \sqrt{3}: 2[/tex] so the hypotenuse, which is the side of the equilateral triangle, is
[tex]s = \dfrac{2}{\sqrt{3}} (4 \sqrt{3}) = 8[/tex]
so an area of a single triangle
[tex] A_\triangle = \frac 1 2 s h = 16 \sqrt{3} [/tex]
The entire hexagon is six of these
[tex] A = 6(16 \sqrt{3}) = 96 \sqrt{3} [/tex]
