The isoperimetric theorem [71, Thm 6.1] guarantees the existence of an integral current $A^{\prime}$ supported in $B_{R}$ such that $\partial A^{\prime}=\partial A$ and for which

$$Mass(A^{\prime})^{(n-1)/n}\leq CMass(\partial A).$$

Notice the metric uniform equivalence means we do not need to be careful to distinguish the Hausdorff measure for the Euclidean metric and the Calabi-Yau metric. Let $T$ denote the cone over the current $A-A^{\prime}$, then $\partial T=A-A^{\prime}$, and thus by the quantitative calibrated condition,

$$\begin{split}&Mass(A)\leq\frac{1}{\sin\epsilon}\int_{A}\text{Re}\Omega=\frac{1}{\sin\epsilon}\int_{A^{\prime}}\text{Re}\Omega+\frac{1}{\sin\epsilon}\int_{\partial T}\text{Re}\Omega\\ \leq&\frac{1}{\sin\epsilon}Mass(A^{\prime})+\frac{1}{\sin\epsilon}\int_{T}d\text{Re}\Omega\leq\frac{1}{\sin\epsilon}CMass(\partial A)^{n/(n-1)},\end{split}$$

which is the isoperimetric inequality. ∎