Question 193354: what is the area of a regular hexagon with a perimeter of 90 centimeters
Answer by RAY100(1637)   (Show Source):
You can put this solution on YOUR website! very complicated,,,please start with a sketch, or refer to geometry book under area polygons
regular hexagon is six sided, with all sides equal.
if we make a rough sketch, and then find center roughly.
from center draw a line to each of vertex,,,,should be six triangles.
the central angle of each triangle should be 120 deg,,,,360/6
now draw an altitude on just one of the triangles,,,,it should go from center to opposite base.
It bisects the base, and is perpendicular.
That smaller triangle should have angles of 90(at base),60(at center, half of 120), and 30(180-60-90)
Remember that a 30 -60-90 triangle have sides proportional to, 1-2-sqrt3
the base side is 90/6=15 long,
the base of the small triangle is half of 15 or 7.5 degrees
this 7.5 side is proportional to the sqrt3 typical side
the height is proportional to 1, on the typical triangle.
(sqrt3/1)=(7.5/h)
cross mult
sqrt3 *h=7.5
h=7.5/sqrt3=4.333
now the area of any triangle is (1/2)base *height
the larger triangle has an area of (1/2) 15*4.33 = 32.475
but a hexagon is six large triangles, 6*32.475=194.85
in general the area of any polygon is 1/2 * perimeter*apothem
apothem (a) is fancy word for the height above, perp distance from base to center
in our case, A =1/2*90*4.33=194.85
normally, in area of polygon problems, the apothem is given
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