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The moduli space of $n$-gons up to orientation preserving similarity can be identified with $\mathbb{C}P^{n-2}$, so I would say $2n-4$.2n-3$. See On the moduli space of polygons in the Euclidean plane by Kapovich and Millson for more about this space. 1 The moduli space of$n$-gons can be identified with$\mathbb{C}P^{n-2}$, so I would say$2n-4\$.