Puzzle: Don't step on the grass

I was suddenly reminded of a geometric puzzle.  This puzzle is not original, but the thematic content is mine.

There is a patch of grass on a high school grounds.  It's shaped like a square (each side being one unit long).  The students aren't supposed to walk on it, but they do.  You could prevent them from walking on the grass by surrounding it with a fence.

But you can save on some fencing by exploiting a key fact about these high school students: they will only ever walk through the grass in straight lines.  So as long as you build enough fences to block all straight lines through the grass, you can prevent students from walking through it.  (Note that paths skimming the edge of the grass don't count, but paths cutting through the corners do count.)  What's the minimum length of fencing necessary?

If that puzzle is too easy, imagine that the grass is shaped like a regular hexagon.

I don't know if this will help, but there's a website which lets you construct geometrical shapes, as if you had a ruler and compass. Here is a square to start, and here is a hexagon.

I would have asked about a grass field shaped like a regular pentagon, but it's quite difficult.  Try it if you're brave.

See the solution