Question 980109
I am making a seven sided regular polygon, each side has a measurement of 30 yd.
What is the total ground area in square yards?
<pre>
Each interior angle using formula {{{((n-2)*"180°")/n}}}{{{""=""}}}{{{((7-2)*"180°")/7}}}{{{128&4/7}}}°.  In the second figure that angle is divided
in half so it's {{{64&2/7}}}°.  The 30 yd is also divided in half, so each
is 15 yd.
 



{{{drawing(400,400,-1.1,1.1,-1.1,1.1,
locate(-.53,-.62,128&4/7),

locate(-.32,-.67,"°"),

red(arc(-0.43388374,-0.90096887,.6,-.6,0,129)),

line(0,1,-0.78183148,0.6234898),
line(-0.78183148,0.6234898,-0.97492791,-0.22252093),
line(-0.97492791,-0.22252093,-0.43388374,-0.90096887),
line(-0.43388374,-0.90096887,0.43388374,-0.90096887),
line(0.43388374,-0.90096887,0.97492791,-0.22252093),
line(0.97492791,-0.22252093,0.78183148,0.6234898),locate(0,-.91,30),
line(0.78183148,0.6234898,0,1))}}}{{{drawing(400,400,-1.1,1.1,-1.1,1.1,

green(line(-0.43388374,-0.90096887,0,0),line(0,0,0,-.90096887), 
line(0.43388374,-0.90096887,0,0)),
locate(-.2,-.91,15),locate(.2,-.91,15),
locate(-.36,-.69,64&2/7),locate(-.21,-.7,"°"),
red(arc(-0.43388374,-0.90096887,.7,-.7,0,65)),
locate(.02,-.45,h),

line(0,1,-0.78183148,0.6234898),
line(-0.78183148,0.6234898,-0.97492791,-0.22252093),
line(-0.97492791,-0.22252093,-0.43388374,-0.90096887),
line(-0.43388374,-0.90096887,0.43388374,-0.90096887),
line(0.43388374,-0.90096887,0.97492791,-0.22252093),
line(0.97492791,-0.22252093,0.78183148,0.6234898),
line(0.78183148,0.6234898,0,1))}}}

By the right triangle on the left,

{{{tan(64&2/7)}}}{{{""=""}}}{{{h/15}}}
{{{15*tan(64&2/7)}}}{{{""=""}}}{{{h}}}

Then the area of each of these 7 isosceles triangles

{{{drawing(400,400,-1.1,1.1,-1.1,1.1,

line(0,1,0,0),
line(-0.78183148,0.6234898,0,0),
line(-0.97492791,-0.22252093,0,0),
line(-0.43388374,-0.90096887,0,0),
line(0.43388374,-0.90096887,0,0),
line(0.97492791,-0.22252093,0,0),
line(0.78183148,0.6234898,0,0),




line(0,1,-0.78183148,0.6234898),
line(-0.78183148,0.6234898,-0.97492791,-0.22252093),
line(-0.97492791,-0.22252093,-0.43388374,-0.90096887),
line(-0.43388374,-0.90096887,0.43388374,-0.90096887),
line(0.43388374,-0.90096887,0.97492791,-0.22252093),
line(0.97492791,-0.22252093,0.78183148,0.6234898),locate(0,-.91,30),
line(0.78183148,0.6234898,0,1))}}}

is found by the formula 

{{{A=expr(1/2)bh=expr(1/2)(30)(15*tan(64&2/7))}}}

We multiply that by 7 and get

{{{7*expr(1/2)(30)(15*tan(64&2/7))}}}{{{""=""}}}{{{matrix(1,3,3270.5212,square,yards)}}}

Edwin</pre>