$A_{g}$ can be explicitly compactified as $\overline{A_{g}}^{\rm GH}$ whose boundary parametrizes all flat (real) tori $\mathbb{R}^{i}/\mathbb{Z}^{i}$ of diameter $1$ where $1\leq i\leq g$. Once we attach the rescaled flat Kähler metric in the principal polarization with diameter $1$ to each abelian variety, the parametrization of metric spaces on whole $\overline{A_{g}}^{\rm GH}$ is continuous with respect to the Gromov-Hausdorff distance.