Consider a degenerating family of elliptic curves parameterized by $t$, given by $\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)$ where $1$ and $\tau={1\over 2\pi i}\log t$ are periods of the elliptic curves. If we take $t$ approaching $0$ along the positive real axis, then we can just view this as a family of elliptic curves $\mathcal{X}_{\alpha}$ with period $1$ and $i\alpha$ with $\alpha\rightarrow\infty$. If we take the standard Euclidean metric $g$ on $\mathcal{X}_{\alpha}$, then the diameter of $\mathcal{X}_{\alpha}$ is unbounded. To obtain a bounded diameter, we replace $g$ by $g/\alpha^{2}$; equivalently, we can keep $g$ fixed on $\mathbb{C}$ but change the periods of the elliptic curve to $1/\alpha,i$. It then becomes clear that the Gromov-Hausdorff limit of such a sequence of elliptic curves is a circle $\mathbb{R}/\mathbb{Z}$.