5. D.

Since the dodecagon and the triangle together have 15 sides, and the two overlapping sides definitely will not be sides of the new convex polygon, the new convex polygon can have at most 13 sides.
It is easy to know that each interior angle of the regular dodecagon paper piece is
$$
\frac{(12-2) \times 180^{\circ}}{12}=150^{\circ} \text {. }
$$

If the triangle $\triangle A B C$ being joined has $B C=1$, then there are the following three scenarios:
(1) $\angle A B C=\angle A C B=90^{\circ}$, the resulting shape after joining is an 11-sided polygon;
(2) $\angle A B C=30^{\circ}, \angle A C B<30^{\circ}$, the resulting shape after joining is a 12-sided polygon;
(3) $\angle A B C<30^{\circ}$, and $\angle A C B<30^{\circ}$, the resulting shape after joining is a 13-sided polygon.

Thus, the number of sides of the convex polygon obtained is definitely not 14.
5. A triangular paper piece with a side length of 1 is placed on a regular 12-sided paper piece with a side length of 1, such that one side of the triangle coincides with one side of the 12-sided polygon, and both are flat on the table. Then the number of sides of the resulting convex polygon must not be $(\quad)$.
(A) 11
(B) 12
(C) 13
(D) 14