Question 766552
<pre>
Here is a different approach.  It's longer but it doesn't
use the formula for the area of a regular polygon,
which is not usually studied in a geometry or trig class.
The following way only uses the interior angle formula,
the tangent equation, and the area of a trapezoid (or 
trapezium) formula.
 
Each interior angle of an n-sided regular polygon is

{{{((n-2)*"180°")/n}}} = {{{((5-2)*"180°")/5}}} = 108° 

{{{drawing(400,400,-2.5,2.5,-2.5,2.5,

line(cos((18+72*0)*pi/180), sin((18+72*0)*pi/180),cos((18+72*1)*pi/180), sin((18+72*1)*pi/180)), 
line(cos((18+72*1)*pi/180), sin((18+72*1)*pi/180),cos((18+72*2)*pi/180), sin((18+72*2)*pi/180)),
line(cos((18+72*2)*pi/180), sin((18+72*2)*pi/180),cos((18+72*3)*pi/180), sin((18+72*3)*pi/180)),
line(cos((18+72*3)*pi/180), sin((18+72*3)*pi/180),cos((18+72*4)*pi/180), sin((18+72*4)*pi/180)),
line(cos((18+72*4)*pi/180), sin((18+72*4)*pi/180),cos((18+72*5)*pi/180), sin((18+72*5)*pi/180)),

line(2cos((18+72*0)*pi/180), 2sin((18+72*0)*pi/180),2cos((18+72*1)*pi/180), 2sin((18+72*1)*pi/180)), 
line(2cos((18+72*1)*pi/180), 2sin((18+72*1)*pi/180),2cos((18+72*2)*pi/180), 2sin((18+72*2)*pi/180)),
line(2cos((18+72*2)*pi/180), 2sin((18+72*2)*pi/180),2cos((18+72*3)*pi/180), 2sin((18+72*3)*pi/180)),
line(2cos((18+72*3)*pi/180), 2sin((18+72*3)*pi/180),2cos((18+72*4)*pi/180), 2sin((18+72*4)*pi/180)),
line(2cos((18+72*4)*pi/180), 2sin((18+72*4)*pi/180),2cos((18+72*5)*pi/180), 2sin((18+72*5)*pi/180)),locate(0,-1.66,20), locate(0,-.83,10),
locate(-1.2,-1.3,"108°"),

arc(2cos((18+72*3)*pi/180), 2sin((18+72*3)*pi/180),1.1,-1.1,0,108)



)}}}{{{matrix(22,1,

We, draw, in, these, 5, lines, and, see, that, the, desired, area, is,
5, isosceles, trapezoids, and, the, "108°",angle, is, bisected)}}}{{{drawing(400,400,-2.5,2.5,-2.5,2.5,

line(cos((18+72*0)*pi/180), sin((18+72*0)*pi/180),cos((18+72*1)*pi/180), sin((18+72*1)*pi/180)), 
line(cos((18+72*1)*pi/180), sin((18+72*1)*pi/180),cos((18+72*2)*pi/180), sin((18+72*2)*pi/180)),
line(cos((18+72*2)*pi/180), sin((18+72*2)*pi/180),cos((18+72*3)*pi/180), sin((18+72*3)*pi/180)),
line(cos((18+72*3)*pi/180), sin((18+72*3)*pi/180),cos((18+72*4)*pi/180), sin((18+72*4)*pi/180)),
line(cos((18+72*4)*pi/180), sin((18+72*4)*pi/180),cos((18+72*5)*pi/180), sin((18+72*5)*pi/180)),

line(2cos((18+72*0)*pi/180), 2sin((18+72*0)*pi/180),2cos((18+72*1)*pi/180), 2sin((18+72*1)*pi/180)), 
line(2cos((18+72*1)*pi/180), 2sin((18+72*1)*pi/180),2cos((18+72*2)*pi/180), 2sin((18+72*2)*pi/180)),
line(2cos((18+72*2)*pi/180), 2sin((18+72*2)*pi/180),2cos((18+72*3)*pi/180), 2sin((18+72*3)*pi/180)),
line(2cos((18+72*3)*pi/180), 2sin((18+72*3)*pi/180),2cos((18+72*4)*pi/180), 2sin((18+72*4)*pi/180)),
line(2cos((18+72*4)*pi/180), 2sin((18+72*4)*pi/180),2cos((18+72*5)*pi/180), 2sin((18+72*5)*pi/180)),

green(
line(2cos((18+72*0)*pi/180),2sin((18+72*0)*pi/180),cos((18+72*0)*pi/180), sin((18+72*0)*pi/180)), 
line(2cos((18+72*1)*pi/180),2sin((18+72*1)*pi/180),cos((18+72*1)*pi/180), sin((18+72*1)*pi/180)),
line(2cos((18+72*2)*pi/180),2sin((18+72*2)*pi/180),cos((18+72*2)*pi/180), sin((18+72*2)*pi/180)),
line(2cos((18+72*3)*pi/180),2sin((18+72*3)*pi/180),cos((18+72*3)*pi/180), sin((18+72*3)*pi/180)),
line(2cos((18+72*4)*pi/180),2sin((18+72*4)*pi/180),cos((18+72*4)*pi/180), sin((18+72*4)*pi/180))),

locate(0,-1.66,20), locate(0,-.83,10),
locate(-.95,-1.35,"54°"),

arc(2cos((18+72*3)*pi/180), 2sin((18+72*3)*pi/180),1.2,-1.2,0,54)



)}}}

Then we draw in two vertical lines in red, which divide
the 20cm side into 5cm, 10cm, and 5cm lengths:

{{{drawing(400,400,-2.5,2.5,-2.5,2.5,

line(cos((18+72*0)*pi/180), sin((18+72*0)*pi/180),cos((18+72*1)*pi/180), sin((18+72*1)*pi/180)), 
line(cos((18+72*1)*pi/180), sin((18+72*1)*pi/180),cos((18+72*2)*pi/180), sin((18+72*2)*pi/180)),
line(cos((18+72*2)*pi/180), sin((18+72*2)*pi/180),cos((18+72*3)*pi/180), sin((18+72*3)*pi/180)),
line(cos((18+72*3)*pi/180), sin((18+72*3)*pi/180),cos((18+72*4)*pi/180), sin((18+72*4)*pi/180)),
line(cos((18+72*4)*pi/180), sin((18+72*4)*pi/180),cos((18+72*5)*pi/180), sin((18+72*5)*pi/180)),

line(2cos((18+72*0)*pi/180), 2sin((18+72*0)*pi/180),2cos((18+72*1)*pi/180), 2sin((18+72*1)*pi/180)), 
line(2cos((18+72*1)*pi/180), 2sin((18+72*1)*pi/180),2cos((18+72*2)*pi/180), 2sin((18+72*2)*pi/180)),
line(2cos((18+72*2)*pi/180), 2sin((18+72*2)*pi/180),2cos((18+72*3)*pi/180), 2sin((18+72*3)*pi/180)),
line(2cos((18+72*3)*pi/180), 2sin((18+72*3)*pi/180),2cos((18+72*4)*pi/180), 2sin((18+72*4)*pi/180)),
line(2cos((18+72*4)*pi/180), 2sin((18+72*4)*pi/180),2cos((18+72*5)*pi/180), 2sin((18+72*5)*pi/180)),

green(
line(2cos((18+72*0)*pi/180),2sin((18+72*0)*pi/180),cos((18+72*0)*pi/180), sin((18+72*0)*pi/180)), 
line(2cos((18+72*1)*pi/180),2sin((18+72*1)*pi/180),cos((18+72*1)*pi/180), sin((18+72*1)*pi/180)),
line(2cos((18+72*2)*pi/180),2sin((18+72*2)*pi/180),cos((18+72*2)*pi/180), sin((18+72*2)*pi/180)),
line(2cos((18+72*3)*pi/180),2sin((18+72*3)*pi/180),cos((18+72*3)*pi/180), sin((18+72*3)*pi/180)),
line(2cos((18+72*4)*pi/180),2sin((18+72*4)*pi/180),cos((18+72*4)*pi/180), sin((18+72*4)*pi/180))),

locate(0,-1.6,10), locate(0,-.83,10),  locate(-.95,-1.6,5),locate(.8,-1.6,5), 
locate(-.95,-1.35,"54°"),locate(-.55,-1.1,h),

arc(2cos((18+72*3)*pi/180), 2sin((18+72*3)*pi/180),1.2,-1.2,0,54),

red(line(cos((18+72*4)*pi/180),2sin((18+72*4)*pi/180),cos((18+72*4)*pi/180), sin((18+72*4)*pi/180)),line(cos((18+72*3)*pi/180),2sin((18+72*3)*pi/180),cos((18+72*3)*pi/180), sin((18+72*3)*pi/180)))



)}}}
In the right triangle, we find h from

tan(54°) = {{{h/5}}}

h = 5·tan(54°)

Area of trapezoid = {{{expr(1/2)(b[1]+b[2])*h}}} = {{{expr(1/2)(10+20)*5*tan("54°")}}}

Area of all 5 trapezoids = {{{5*expr(1/2)(10+20)*5*tan("54°")}}} = 516.1432202 cm²

Edwin</pre>