Draw out your regular hexagon.
Let’s find the distance between the top and bottom sides.
That distance can be found in many different ways.
Let’s try to find it by using the symmetry of the regular hexagon.
The distance from the top side to the bottom side is
The length of any perpendicular between those sides.
So imagine a point T on the top side, and then drawing a line perpendicular to the top through T, and extending that line down to the bottom side so that it intersects at point B of the bottom side.
Then TB is the distance we’re looking for.
But notice this length is the same no matter what point T we choose on the top side.
We could choose the leftmost endpoint, the center point, a random point, or the rightmost endpoint of the top side and we’d find the same length, since the top and bottom sides are parallel.
So let’s try choosing T to be the midpoint of the top side. So B is the midpoint of the bottom side.
And we want to find the length of TB
But now note that this segment TB passed through the center O of the hexagon.So all we need to do is find TO and then double that to get TB.
We suspect that we’ll want to use the top side’s length somewhere to find TO,So let’s call the endpoints of the top side A and B.Then notice that ATO forms a right triangle at angle T, since TO is perpendicular to TA.And we know TA = 6/2 = 3, since T is the midpoint of AB.
So we have a right triangle and we know one side length. All we need to find any other side length is an angle.Actually, this is not so bad, since we know that there is symmetry going on.Look at the angle TOA. What is it’s angle? Think about the symmetry of the hexagon, imagine connecting the center O to all other corners of the hexagon, and remember that the sum of the angles around O should equal 360 degrees.
Once you know the angle TOA, and once you know that TA = 3, then it follows thatTA / TO = tan (TOA), so TA / tan (TOA) = TO.
Then double it to find your length TB.
