Trigonometry · Superchallenge problem

A regular hexagon with side length $2$ stands on one side on a flat floor. It is tipped over to the right, pivoting about its bottom-right vertex, until the next side lies on the floor; this is repeated until the hexagon has made one full revolution (six tips) and every side has touched the floor once. Vertex $A$ starts as the bottom-left vertex. Find the length of the path traced by $A$.