A regular hexagon can be split into 6 equilateral triangles of side a where a is the length of each side of the hexagon. The apothem is the vertex angle bisector of any of the triangles, and this angle bisector also bisects the base (a side of the hexagon). This creates a right triangle with a/2 as the length of the bisected base. The angle opposite the bisected base is 30° (half the vertex angle). If the length of the apothem is x, then:
a/(2x)=tan(30)=1/√3, a=2x/√3, which can be written a=2x√3/3. This relates the length of the side of the hexagon to the length of the apothem. The perimeter is 6a=4x√3.
Another way to solve this is to use the fact that the radius also has length a, so the bisected base has length a/2, the radial hypotenuse has length a, so x=√(a2-a2/4)=a√3/2, so a=2x/√3, as before. Perimeter 6a=12x/√3=4x√3.