so.. notice the picture below, the triangle on the far-right is the 30-60-90 rule, those are the ratios
now... since we know the apothem value... using the ratios we end up with 2=x.... so... since now we know the length of that one side on that triangle, we know the hexagon has sides that are all each 2+2, or 4
now, bear in mind, a regular hexagon, has 6 equal sides, the angle from one corner to the other, is there therefore 6 times, now, if you divide 360° of a circle by 6 even angles, you get 60° each "central" angle, as shown in the picture
now, you run the apothem line through that angle, and is an angle bisector, cuts it in half, and you end up with a 30-60-90 triangle
well, now, you know the apothem length and the length of one side, if one side is 4, the perimeter is just 6 * 4 or 24, and now you know the perimeter
[tex]\bf \textit{area of a regular polygon}\\\\
A=\cfrac{1}{2}(apothem)(perimeter)[/tex]
