Fundamental group of polygon, application of van Kampen I have the following figure, $a^6$. I am still not clear on how to use van Kampen's theorem, what are my open sets, (how to determine) and how to use the formula. 
 A: Take a disc in the center and the rest of the polygon minus the center of the disc as open sets. The intersection is an annulus and you can retract the second open set to the boundary of the polygon. Doing this you end up with:
$$\pi_1(X)=L(a)/<<a^6=e>> \cong \mathbb{Z}_6$$
A: In general, if $f : S^1 \to X$ is a loop, then given the mapping cone $C_f = D^2 \sqcup_f X$ obtained from gluing a disk to $X$ by attaching the boundary by $f$, $\pi_1(C_f) \cong \pi_1(X)/\langle [f] \rangle$.
Intuitively speaking, the loop $f$ can be nullhomotoped by sliding through the glued disk, hence we add the relator $[f] = 1$ to the fundamental group. 
Rigorously, use the trick Adolfo used. Pick a disk in the interior of the attached disk in $D^2 \sqcup_f X$ and take the complement. Thicken both to get an open cover $\{U, V\}$, where $U$ is an open disk hence contractible and $V$ deformation retracts to $X$. $U \cap V$ is just the loop represented by $[f]$. From Siefert-van Kampen theorem it now follows that $\pi_1(C_f) \cong \pi_1(X)/\langle [f] \rangle$.
In particular, note that your identification space is the same as $C_f$ where $f: S^1 \to S^1$ is the degree $6$ map $z \mapsto z^6$. Using the previous theorem, $\pi_1(C_f) \cong \langle a| a^6 = 1\rangle \cong \Bbb Z/6$.
