Question 1201660
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Answer:  <font color=red>209.646864 square meters</font> (approximate)



Explanation:


A regular polygon has equal angles and equal sides.
A nonagon has 9 sides.


Let's draw out a regular nonagon.
{{{
drawing(400,400,-4.94,4.94,-4.76,5,
line(0,4,-2.57,3.06),line(-2.57,3.06,-3.94,0.69),line(-3.94,0.69,-3.46,-2),line(-3.46,-2,-1.37,-3.76),line(-1.37,-3.76,1.37,-3.76),line(1.37,-3.76,3.46,-2),line(3.46,-2,3.94,0.69),line(3.94,0.69,2.57,3.06),line(2.57,3.06,0,4),line(0,4,0,4)
)

}}}


If we connected the center to each vertex, then we get 9 congruent mirror copies of the same isosceles triangle.
Think of them as slices of pizza.
{{{
drawing(400,400,-4.94,4.94,-4.76,5,
line(0,4,-2.57,3.06),line(-2.57,3.06,-3.94,0.69),line(-3.94,0.69,-3.46,-2),line(-3.46,-2,-1.37,-3.76),line(-1.37,-3.76,1.37,-3.76),line(1.37,-3.76,3.46,-2),line(3.46,-2,3.94,0.69),line(3.94,0.69,2.57,3.06),line(2.57,3.06,0,4),


line(0,0,-0.0000,4.0000),
line(0,0,-2.5712,3.0642),
line(0,0,-3.9392,0.6946),
line(0,0,-3.4641,-2.0000),
line(0,0,-1.3681,-3.7588),
line(0,0,1.3681,-3.7588),
line(0,0,3.4641,-2.0000),
line(0,0,3.9392,0.6946),
line(0,0,2.5712,3.0642),
line(0,0,0,4.0000)

)

}}}
Divide 360 degrees over 9 equal pieces.
360/9 = 40



The vertex angle of each isosceles triangle is 40 degrees.
{{{
drawing(400,400,-4.94,4.94,-4.76,5,
line(0,4,-2.57,3.06),line(-2.57,3.06,-3.94,0.69),line(-3.94,0.69,-3.46,-2),line(-3.46,-2,-1.37,-3.76),line(-1.37,-3.76,1.37,-3.76),line(1.37,-3.76,3.46,-2),line(3.46,-2,3.94,0.69),line(3.94,0.69,2.57,3.06),line(2.57,3.06,0,4),


line(0,0,-0.0000,4.0000),
line(0,0,-2.5712,3.0642),
line(0,0,-3.9392,0.6946),
line(0,0,-3.4641,-2.0000),
line(0,0,-1.3681,-3.7588),
line(0,0,1.3681,-3.7588),
line(0,0,3.4641,-2.0000),
line(0,0,3.9392,0.6946),
line(0,0,2.5712,3.0642),
line(0,0,0,4.0000),

red(
locate(-0.2,-1,40^o),
arc(0,0,2,2,70,110)
)

)

}}}




Draw a vertical line through that triangle at the bottom.
This splits the 40 degree angle into two 20 degree pieces.

{{{
drawing(400,400,-2,2,-4.5,1,

line(-1.37,-3.76,1.37,-3.76),
line(0,0,-1.3681,-3.7588),
line(0,0,1.3681,-3.7588),
locate(0.1,-3,8),

blue(
locate(-0.3,-1,20^o),
arc(0,0,2,2,90,110)
),

red(
locate(-0.5,-2,40^o),
arc(0,0,4,4,70,110)
),

line(0,0,0,-0.09397),line(0,-0.18794,0,-0.28191),line(0,-0.37588,0,-0.46985),line(0,-0.56382,0,-0.65779),line(0,-0.75176,0,-0.84573),line(0,-0.9397,0,-1.03367),line(0,-1.12764,0,-1.22161),line(0,-1.31558,0,-1.40955),line(0,-1.50352,0,-1.59749),line(0,-1.69146,0,-1.78543),line(0,-1.8794,0,-1.97337),line(0,-2.06734,0,-2.16131),line(0,-2.25528,0,-2.34925),line(0,-2.44322,0,-2.53719),line(0,-2.63116,0,-2.72513),line(0,-2.8191,0,-2.91307),line(0,-3.00704,0,-3.10101),line(0,-3.19498,0,-3.28895),line(0,-3.38292,0,-3.47689),line(0,-3.57086,0,-3.66483)
)
}}}
The apothem is represented as the dashed line.
It is perpendicular to the polygon's edge.


Focus on half of the triangle.
{{{
drawing(400,400,-2,2,-4.5,1,
line(0,0,0,-3.7588),
line(0,0,0,-3.7588),
line(0,0,-1.3681,-3.7588),

line(-0.2,-3.7588,-0.2,-3.7588+0.2),
line(-0.2,-3.7588+0.2,0,-3.7588+0.2),


line(-1.3681,-3.7588,0,-3.7588),
locate(0.1,-2,8),
locate(-0.75,-3.8,x),
locate(-0.3,-1,20^o)
)
}}}


We'll use the tangent ratio to determine x.


tan(angle) = opposite/adjacent
tan(20) = x/8
x = 8*tan(20)
x = 2.911762 approximately
Your calculator needs to be in degree mode.


Double this value to determine the approximate side length of this regular nonagon.
2x = 2*2.911762 = 5.823524


{{{
drawing(400,400,-4.94,4.94,-5.16,4.6,
line(0,4,-2.57,3.06),line(-2.57,3.06,-3.94,0.69),line(-3.94,0.69,-3.46,-2),line(-3.46,-2,-1.37,-3.76),line(-1.37,-3.76,1.37,-3.76),line(1.37,-3.76,3.46,-2),line(3.46,-2,3.94,0.69),line(3.94,0.69,2.57,3.06),line(2.57,3.06,0,4),


line(0,0,-0.0000,4.0000),
line(0,0,-2.5712,3.0642),
line(0,0,-3.9392,0.6946),
line(0,0,-3.4641,-2.0000),
line(0,0,-1.3681,-3.7588),
line(0,0,1.3681,-3.7588),
line(0,0,3.4641,-2.0000),
line(0,0,3.9392,0.6946),
line(0,0,2.5712,3.0642),
line(0,0,0,4.0000),


locate(0.1,-2,8),
locate(-0.8,-4.8,5.823524),

line(0,0,0,-0.09397),line(0,-0.18794,0,-0.28191),line(0,-0.37588,0,-0.46985),line(0,-0.56382,0,-0.65779),line(0,-0.75176,0,-0.84573),line(0,-0.9397,0,-1.03367),line(0,-1.12764,0,-1.22161),line(0,-1.31558,0,-1.40955),line(0,-1.50352,0,-1.59749),line(0,-1.69146,0,-1.78543),line(0,-1.8794,0,-1.97337),line(0,-2.06734,0,-2.16131),line(0,-2.25528,0,-2.34925),line(0,-2.44322,0,-2.53719),line(0,-2.63116,0,-2.72513),line(0,-2.8191,0,-2.91307),line(0,-3.00704,0,-3.10101),line(0,-3.19498,0,-3.28895),line(0,-3.38292,0,-3.47689),line(0,-3.57086,0,-3.66483),

arc(-1.096,-4.034,0.548,0.548,90,180),line(-1.096,-4.308,-0.274,-4.308),arc(-0.274,-4.582,0.548,0.548,270,360),arc(0.274,-4.582,0.548,0.548,180,270),line(0.274,-4.308,1.096,-4.308),arc(1.096,-4.034,0.548,0.548,0,90)

)

}}}


area of one triangle = 0.5*base*height
area of one triangle = 0.5*5.823524*8
area of one triangle = 23.294096


area of nonagon = area of nine triangles 
area of nonagon = 9*(area of one triangle)
area of nonagon = 9*(23.294096)
area of nonagon = 9*(23.294096)
area of nonagon = <font color=red>209.646864 square meters</font>

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