The apothem is the long leg of a 30-60-90 right triangle that has as its short side half the length of one side of the hexagon. That half-side length is (3√3)/√3 = 3. The area of the figure is given by the formula A = (1/2)Pa where p is the perimeter (number of sides × side length) and "a" is the apothem.
Filling in the numbers, you have A = (1/2)(6·6 mm)(3√3 mm) = 54√3 mm²
so, if we inscribe that regular hexagon in a circle, the sides of the hexagon will make an equilateral triangle with the center of the circle and two radii, check the picture below.
so running a line from the center to a vertex of the hexagon and linking it with the apothem line, which is an angle bisector, we end up with a 30-60-90 triangle, as you see there in the picture, so we can use the 30-60-90 rule to get the side "s".
so, now we know half of a side is 3, that means one side of the hexagon is 3+3 or 6, well, there are 6 sides, so the perimeter of the hexagon is 6+6+6+6+6+6, namely 36.
[tex]\bf \textit{area of a regular polygon}\\\\
A=\cfrac{1}{2}ap~~
\begin{cases}
a=apothem\\
p=perimeter\\
-------\\
a=3\sqrt{3}\\
p=36
\end{cases}\implies A=\cfrac{1}{2}(3\sqrt{3})(36)
\\\\\\
A=54\sqrt{3}[/tex]
