  We study the function $$\mbox{KVol} : (X,\omega)\mapsto \mbox{Vol} (X,\omega)
\sup_{\alpha,\beta} \frac{\mbox{Int} (\alpha,\beta)}{l_g (\alpha) l_g
(\beta)}$$
  defined on the moduli spaces of translation surfaces. More precisely, let
$\mathcal T_n$ be the Teichm\"uller discs of the original Veech surface
$(X_n,\omega_n)$ arising from right-angled triangle with angles
$(\pi/2,\pi/n,(n-2)\pi/2n)$ by the unfolding construction for $n\geq 5$. For $n
\equiv 1 \mod 2$ and any $(X,\omega)\in \mathcal T_n$, we establish the (sharp)
bounds $$ \frac{n}{2} \cot \frac{\pi}{n} \leq \mbox{KVol}(X,\omega) \leq
\frac{n}{2} \cot \frac{\pi}{n} \cdot \frac1{\sin \frac{2\pi}{n}}.$$
  The lower bound is uniquely realized at $(X_n,\omega_n)$.
