Drawing lines from the centre to each vertex of the hexagon, as shown, splits it into 6 identical isosceles triangles.
The total angle at the centre is $360^\text o$, so the angle at the centre of each triangle must be $360\div6 = 60^\text o$.
Using trigonometry
Since the angle at the middle of each triangle is $60^\text o$, when the triangle is split in half by the radius, as shown below, the angle will be $30^\text o$. In the diagram, $s$ denotes the side lengthof the hexagon.
$\tan {30^\text o}=\dfrac{\frac{s}{2}}{1}\rightarrow\tan{30^\text o}=\frac{s}{2}\rightarrow2\tan{30^\text o}=s$.
$\tan{30^\text o}=\dfrac{1}{\sqrt3}$, so $s=2\times\dfrac{1}{\sqrt3}=\dfrac{2}{\sqrt{3}}$ or $1.155$.
The hexagon has $6$ sides, so its perimeter is $6\times\dfrac{2}{\sqrt3}=2\times\sqrt3\times\sqrt3\times\dfrac{2}{\sqrt3}=4\sqrt3$, or $6\times1.155=6.93$.
Using Pythagoras' Theorem
Since the angle at the middle of each triangle is $60^\text o$, the triangles must be equilateral triangles (as they are already isosceles, so the other two angles are equal, and add up to $120^\text o$).
The diagram below shows one of the triangles cut in half by the radius, where the side length of the hexagon is labelled $s$.
The hexagon has $6$ sides, so its perimeter is $6\times\dfrac{2}{\sqrt3}=2\times\sqrt3\times\sqrt3\times\dfrac{2}{\sqrt3}=4\sqrt3$, or $6\times1.155=6.93$.