A regular polygon with 30 vertices

Let $P$ be the $30$ sided polygon. There are a total of $\binom{30}{3}=4060$ triangles we can form using the vertices of $P$ . Some of these share sides with $P$ , so we need to subtract these.

Exactly one side in common

One side of the triangle is a side of $P$ . There are $30$ possible sides. The remaining vertex of the triangle is not on this side, and is not adjacent to it; there are $26$ such points, for a total of $26 \times 30=780$ triangles sharing exactly one side with $P$ .

Exactly two sides in common

The vertices of these triangles are three consecutive vertices of $P$ . Counting the middle one of these, there are $30$ such triangles.

---

Hence there are $4060-780-30=\boxed{3250}$ triangles that do not share a side with $P$ .