Question 193049: what is the area of a regular hexagon whose sides are each 12 inches long. Round to the nearest squre inch. I drew the figure. used this formula: A=4 + 2, A=6. Or should it be 360
Found 2 solutions by Alan3354, RAY100:
Answer by Alan3354(69443)   (Show Source):
You can put this solution on YOUR website! what is the area of a regular hexagon whose sides are each 12 inches long. Round to the nearest squre inch. I drew the figure. used this formula: A=4 + 2, A=6. Or should it be 360
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If you draw lines from the center to each vertex, the hexagon is made up of 6 equilateral triangles, 12" on a side.
The height of the triangle is sqrt(12^2 - 6^2) = sqrt(108)
The area of one triangle is bh/2 = 12*sqrt(108)/2 = 6sqrt(108)
There are 6, so the total area is 36sqrt(108)
= 216sqrt(3)
= ~ 62 sq inches
Answer by RAY100(1637)  (Show Source):
You can put this solution on YOUR website! good start,,somewhat complex solution,,your geometry book probably has a good ref
lets start with the sketch,,,regular hexagon has 6 equal sides,12 for this problem.
If we find the center of the hexagon, and then draw lines to each vertex, we have 6 identical triangles.
Area of a triangle =1/2 base * height
ok, but we only know base, we need to find height
We also know the central angle,,,,,remember sum of angles of polygon =(n-2)(180)
in our case (6-2)(180)=720,,,,,with 6 angles,,,720/6=120,,,(central angle of triangle).
note the 120 deg on one triangle.
also , please, draw an altitude on that triangle, from center to base.
this is the height of that triangle.
Note the altitude bisects the 120 angle into 2 equal 60 deg angles.
we are almost there,
in our triangle we have the 60 degree angle at the center vertex, 90 deg where altitude intersects the base, therefore the other vertex angle is 30 degrees,,,(sum of triangle =180 degrees)
lets mark the 30 degree vertex, and the 90 degree angle.
we now have a 30-60-90 degree triangle, with one side (opposite the 60 angle) equal to 6
which is half of the full side 12.
we know from typical triangles that a 30-60-90 has sides of 1,2,sqrt3.
in our case we need height,h, which is opposite the 30 deg angle.
if the height is proportional to (1), hyp is (2), and the base side (sqrt3),,,but base side is 6
sqrt3/6=1/h
cross multiply,,
(sqrt3)(h)=6
h=6/sqrt3=3.464
now,,,area of triangle is 1/2 (base) (height) = 1/2 (12)(3.464)=20.78,,,,(full base=12)
and remember we have 6 triangles in hexagon
area if hexagon=6(20.78)=124.7=125
this is generalized to Area of any polygon =1/2 (perimeter)(apothem)
in our case,,,per =6*size of side, apothem is height
Area=1/2 *(12*6)*3.464= 125 ok
note, apothem is denoted (a), and is usually given,,just remember 1/2 * per*apothem
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