Question 1180960
<pre>
{{{drawing(400,400,-1.1,1.1,-1.1,1.1,

circle(0,0,1),
line(0,1,0.58778525,-0.80901699),
locate(0,0,O),locate(.23,.43,P),


line(-0.95105652,0.30901699,0.58778525,-0.80901699),
line(-0.95105652,0.30901699,0.95105652,0.30901699),

red(line(0,0,0.95105652,0.30901699),line(0,0,.2245139894,0.30901699)), 

locate(1,.37,Q),

line(-0.58778525,-0.80901699,0,1),line(-0.58778525,-0.80901699,0.95105652,0.30901699)) }}}

By sketching in a few other lines, it shouldn't take you long
to figure out that the star is made up of 10 triangles
all congruent to ΔOPQ.  Also, you can easily get:

∠POQ = 36<sup>o</sup>, OQ = 20, ∠OQP = 18<sup>o</sup>, ∠OPQ = 126<sup>o</sup>

By the law of sines,

{{{OQ/sin(OPQ)=PQ/sin(POQ)}}}

{{{20/sin(126^o)=PQ/sin(36^o)}}}

Solve that and get

PQ = 14.53085056

Then to find the area of that triangle, we use the
formula:

{{{Area}}}{{{""=""}}}{{{expr(1/2)OQ*PQ*sin(OQP)}}}

{{{Area}}}{{{""=""}}}{{{expr(1/2)20*14.53085056*sin(18^o)}}}

{{{Area}}}{{{""=""}}}{{{44.90279766}}}{{{cm^2}}}

We multiply that by 10 and get

449.0279766 cm<sup>2</sup>    <---answer

Edwin</pre>