  We consider an embedding of planar maps into an equilateral triangle $\Delta$
which we call the Cardy embedding. The embedding is a discrete approximation of
a conformal map based on percolation observables that are used in Smirnov's
proof of Cardy's formula. Under the Cardy embedding, the planar map induces a
metric and an area measure on $\Delta$ and a boundary measure on $\partial
\Delta$. We prove that for uniformly sampled triangulations, the metric and the
measures converge jointly in the scaling limit to the Brownian disk conformally
embedded into $\Delta$ (i.e., to the $\sqrt{8/3}$-Liouville quantum gravity
disk). As part of our proof, we prove scaling limit results for critical site
percolation on the uniform triangulations, in a quenched sense. In particular,
we establish the scaling limit of the percolation crossing probability for a
uniformly sampled triangulation with four boundary marked points.
