Age 16 to 18 Challenge Level:
The angles are all multiples of $36^o$ and the figure has many
symmetries. Form a quadratic equation by equating ratios of
corresponding sides in the similar triangles.
In explaining the construction look for $\sqrt 5$ appearing from
triangle $YOS$ and the golden ratio appearing as the radius of
circle $C_5$ (see Note). Concentrate on cosines which, in this
case, are easier than the sines to find exactly.