The governor of Reniets (a land far, far away!) is known to be very stingy.
He has to build a road which connects all the five cities in the region, which are oddly arranged on the vertices of a regular pentagon (see the picture below).
Building a road is very expensive: the longer it is, the more it costs!
His first idea is to build five roads, each connecting a city to the center of the pentagon, then he realizes that it might not be optimal. In fact, he remembers that, years ago, his friend had to connect four towers, and the solution wasn't trivial!.
How long is the shortest path that connects all the five cities?
- assume that the side is long $1$ km.
- I currently don't know the solution, I will accept the most convincing answer if no proof is given after some days.
- If you can generalize the solution to polygons of any number $n$ of sides, I will award a bounty (the amount depends on the quality of the work).