A solid diagram of volume n is a packing of n unit cubes into a corner so that the heights of vertical stacks of cubes do not increase in either of two horizontal directions away from the corner. An asymptotic distribution of the dimensions – heights, depths, and widths – of the diagram chosen uniformly at random among all such diagrams is studied. For each k, the planar base of k tallest stacks is shown to be Plancherel distributed in the limit $n\to\infty$.