6. The number of triangles with different shapes that can be formed using the vertices of a regular tridecagon is $\qquad$ (Note: Congruent triangles are considered to have the same shape).
6. 14 .

The vertices of a regular 13-sided polygon divide its circumscribed circle into 13 equal parts. Let the number of parts (in the counterclockwise direction) between the three vertices of a triangle be $a$, $b$, and $c$. Then
$$
a, b, c \in \mathbf{Z}_{+} \text{, and } a+b+c=13 \text{. }
$$

To determine the number of triangles of different shapes, we only need to determine how many possible values the array $(a, b)$ can take.
To ensure that the resulting triangles are all of different shapes, we can set $a \leqslant b \leqslant c$.
Then $1 \leqslant a \leqslant 4$, and $a \leqslant b \leqslant c=13-a-b$
$\Rightarrow a \leqslant b \leqslant \frac{13-a}{2}$.
When $a=1$, $b$ has six possible values;
When $a=2$, $b$ has four possible values;
When $a=3$, $b$ has three possible values;
When $a=4$, $b$ has one possible value.
Therefore, the number of triangles of different shapes with vertices at the vertices of a regular 13-sided polygon is $6+4+3+1=14$.