  We study the isoperimetric subgraphs of the giant component $\textbf{C}_n$ of
supercritical bond percolation on the square lattice. These are subgraphs of
$\textbf{C}_n$ having minimal edge boundary to volume ratio. In contrast to the
work of Biskup, Louidor, Procaccia and Rosenthal, the edge boundary is taken
only within $\textbf{C}_n$ instead of the full infinite cluster. The
isoperimetric subgraphs are shown to converge almost surely, after rescaling,
to the collection of optimizers of a continuum isoperimetric problem emerging
naturally from the model. We also show that the Cheeger constant of
$\textbf{C}_n$ scales to a deterministic constant, which is itself an
isoperimetric ratio, settling a conjecture of Benjamini in dimension two.
