Question 262298
<pre><font size = 4 color = "indigo"><b>
If you are given the regular hexagon:

{{{drawing(200,200,-1.2,1.2,-1.2,1.2,

line(1,0,1/2,sqrt(3)/2),
line(1/2,sqrt(3)/2,-1/2,sqrt(3)/2),
line(-1,0,-1/2,sqrt(3)/2),
line(-1,0,-1/2,-sqrt(3)/2),
line(-1/2,-sqrt(3)/2,1/2,-sqrt(3)/2),
line(1,0,1/2,-sqrt(3)/2) )}}}

Draw in all the "spokes":

{{{drawing(200,200,-1.2,1.2,-1.2,1.2,

line(1,0,1/2,sqrt(3)/2),
line(1/2,sqrt(3)/2,-1/2,sqrt(3)/2),
line(-1,0,-1/2,sqrt(3)/2),
line(-1,0,-1/2,-sqrt(3)/2),
line(-1/2,-sqrt(3)/2,1/2,-sqrt(3)/2),
line(1,0,1/2,-sqrt(3)/2),

line(0,0,1/2,sqrt(3)/2),
line(-1/2,sqrt(3)/2,0,0),
line(-1,0,0,0),
line(0,0,1/2,-sqrt(3)/2),
line(-1/2,-sqrt(3)/2,0,0),
line(1,0,0,0)
 )}}} 

All 6 triangles are congruent equilateral triangles.
Find the area of just the bottom triangle, the green one,
by calculating its height, the red line, by the Pythagorean 
theorem.

{{{drawing(200,200,-1.2,1.2,-1.2,1.2,

line(1,0,1/2,sqrt(3)/2),
line(1/2,sqrt(3)/2,-1/2,sqrt(3)/2),
line(-1,0,-1/2,sqrt(3)/2),
line(-1,0,-1/2,-sqrt(3)/2),
green(line(-1/2,-sqrt(3)/2,1/2,-sqrt(3)/2)),
line(1,0,1/2,-sqrt(3)/2),

line(0,0,1/2,sqrt(3)/2),
line(-1/2,sqrt(3)/2,0,0),
line(-1,0,0,0), red(line(0,0,0,-sqrt(3)/2)),
green(line(0,0,1/2,-sqrt(3)/2)),
green(line(-1/2,-sqrt(3)/2,0,0)),
line(1,0,0,0)
 )}}} 

Then multiply the area of that one triangle by 6,
and you'll have the area of the whole regular hexagon.

Edwin</pre>