SOLUTION: Hi! I'm trying to figure out a problem with a hexagon. I need to find the side of a regular hexagon if the it is 18 inches apart from midpoint to midpoint. Any help would be greatl

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Question 42824: Hi! I'm trying to figure out a problem with a hexagon. I need to find the side of a regular hexagon if the it is 18 inches apart from midpoint to midpoint. Any help would be greatly appreciated!
Thankyou!
Found 2 solutions by fractalier, psbhowmick:
Answer by fractalier(2101) About Me  (Show Source):
You can put this solution on YOUR website!
You can use trig to do this...
Make a triangle with the segment connecting the midpoints of the sides as the base...
Then bisect the top angle and drop an altitude, making two right triangles...
Each right triangle has a top angle of 67.5 degrees...the long leg is 9, which is half of the 18...the hypotenuse is half the side of the hexagon...
Thus sin 67.5 = 9/x and
x = 9 / (sin 67.5) = 9.74
and the side is double that, or about
19.48 inches

Answer by psbhowmick(489) About Me  (Show Source):
You can put this solution on YOUR website!
Your statement of the problem is not very clear.
I assume, what you meant is: The side of a regular hexagon in which the distance between the opposite sides is 18".

See the regular hexagon ABCDEF below.
drawing%28400%2C400%2C-10%2C10%2C-10%2C10%2C%0D%0Aline%286%2C0%2C3%2C5.2%29%2C%0D%0Aline%286%2C0%2C3%2C-5.2%29%2C%0D%0Aline%28-6%2C0%2C-3%2C5.2%29%2C%0D%0Aline%28-6%2C0%2C-3%2C-5.2%29%2C%0D%0Aline%283%2C5.2%2C-3%2C5.2%29%2C%0D%0Aline%283%2C-5.2%2C-3%2C-5.2%29%2C%0D%0Alocate%286%2C0%2CA%29%2C%0D%0Alocate%283%2C5.2%2CB%29%2C%0D%0Alocate%28-3%2C5.2%2CC%29%2C%0D%0Alocate%28-6%2C0%2CD%29%2C%0D%0Alocate%28-3%2C-5.2%2CE%29%2C%0D%0Alocate%283%2C-5.2%2CF%29%0D%0A%29

According to the problem the distance between sides AB & DE, BC & EF and CD & FA is 18" and we are required to find length AB or any other side.

Join B and F with a straight line. Then drop a perpendicular to BF from A intersecting BF at G.

drawing%28400%2C400%2C-10%2C10%2C-10%2C10%2C%0D%0Aline%286%2C0%2C3%2C5.2%29%2C%0D%0Aline%286%2C0%2C3%2C-5.2%29%2C%0D%0Aline%28-6%2C0%2C-3%2C5.2%29%2C%0D%0Aline%28-6%2C0%2C-3%2C-5.2%29%2C%0D%0Aline%283%2C5.2%2C-3%2C5.2%29%2C%0D%0Aline%283%2C-5.2%2C-3%2C-5.2%29%2C%0D%0Alocate%286%2C0%2CA%29%2C%0D%0Alocate%283%2C5.2%2CB%29%2C%0D%0Alocate%28-3%2C5.2%2CC%29%2C%0D%0Alocate%28-6%2C0%2CD%29%2C%0D%0Alocate%28-3%2C-5.2%2CE%29%2C%0D%0Alocate%283%2C-5.2%2CF%29%2C%0D%0Aline%283%2C5.2%2C3%2C-5.2%29%2C%0D%0Aline%286%2C0%2C3%2C0%29%2C%0D%0Alocate%283%2C0%2CG%29%0D%0A%29

Now, in triangle ABF,
BF = 18",
< BAF = 120%5Eo [since each internal angle of a regular hexagon is 120%5Eo]

Clearly, triangle ABG and triangle AFG are congruent.
So, < BAG = 1%2F2< BAF = 60%5Eo and BG = 1%2F2BF = 9".

Now, in triangle ABG,
< AGB = 90%5Eo [since AG is perpendicular to BF]
So in right angled triangle AGB,
sin(< BAG) = BG%2FAB
or sin60%5Eo = 9%2FAB
or AB = 9%2Fsin%2860%5Eo%29" = 9%2F%28sqrt%283%29%2F2%29" = 6sqrt%283%29" = 10.4" (approx)

Thus the reqd. side of the regular hexagon is approximately 10.4 inches.