The classical version of the van Kampen theorem is concerned about the fundamental group of a based space. In fact, it says that the functor $\pi\_1$ preserves certain types of pushouts in $Top\_\*$.

There is also a generalization of the van Kampen theorem that holds for the fundamental groupoid of a space $X$, where in this case it states precisely that the fundamental groupoid functor, $\Pi$, preserves certain colimits in $Top$, namely, those that arise from "nice" open coverings of $X$.

The "groupoid" version of the van Kampen theorem seems to me more conceptual and more elegant than the classical version. Also, the groupoid version allows one to prove the classical version in a more or less easier way. 

Although, apart from the conceptual advantages of the groupoid version of van Kampen theorem, I would like to know if we are able to do any interesting calculations using the fundamental groupoid version of the van Kampen. In fact, explicitly describing a groupoid as a colimit of "simpler" groupoids is something that it is not clear at all for me. I would like to know some concrete cases where is it possible to describe the fundamental groupoid of a space, using this generalization form of the van Kampen theorem, and, if possible, to calculate the fundamental group directly from our calculation of $\Pi(X).$