7. Given a regular $n$-sided polygon ($n$ being an odd number) inscribed in a unit circle, let $S$ be the set of distances between any two distinct vertices of the regular $n$-sided polygon. Then the value of $P=\prod_{i \in S} i$ is $\qquad$ .
7. $\sqrt{n}$.

Let $n=2 m+1\left(m \in \mathbf{N}_{+}\right)$, and denote the polygon as $P_{0} P_{1} \cdots P_{n-1}$. Place the regular polygon in the complex plane with the center at the origin, and the vertices correspond to the complex numbers
$$
P_{k}=\omega^{k}\left(\omega=\mathrm{e}^{\frac{2 \pi i}{n}}\right) .
$$

Let $d_{i}$ be the $i$-th largest distance in $S$. Then
$$
\begin{array}{c}
d_{k}^{2}=\left|P_{0} P_{k}\right|^{2}=\left|1-\omega^{k}\right|^{2} \\
=\left(1-\omega^{k}\right)\left(1-\omega^{-k}\right) \\
=\left(1-\omega^{k}\right)\left(1-\omega^{n-k}\right) .
\end{array}
$$

Then $P^{2}=\prod_{k=1}^{m} d_{k}^{2}=\prod_{k=1}^{n-1}\left(1-\omega^{k}\right)$.
$$
\begin{array}{l}
\text { By } \prod_{k=1}^{n-1}\left(x-\omega^{k}\right)=\frac{x^{n}-1}{x-1}=\sum_{k=0}^{n-1} x^{k} \\
\Rightarrow P=\sqrt{n} .
\end{array}
$$