2008 Cayley Problem 24

A cube has edges of length 1 cm and has a dot marked in the centre of the top face. The cube is sitting on a flat table. The cube is rolled, without lifting or slipping, in one direction so that at least two of its vertices are always touching the table. The cube is rolled until the dot is again on the top face. The length, in centimetres, of the path travelled by the dot is

\textbf{(A)}\ \pi\quad \textbf{(B)}\ 2\pi\quad \textbf{(C)}\ \sqrt{2}\pi\quad \textbf{(D)}\ \sqrt{5}\pi\quad \textbf{(E)}\ \left(\frac{1+\sqrt{5}}{2}\right)\pi

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Cayley contests here.