We apply stochastic geometry to the transition from hexagonal to square cells recently observed in surface-tension-driven Bénard convection. In particular we study the metric and topological evolution of Bénard patterns as a function of the temperature difference, [Formula Presented] across the layer. The preference of square Bénard cells at higher [Formula Presented] is a consequence of both a higher efficiency in heat transfer and more favorable metric properties. Most notably, the perimeter-area ratio of a square cell exceeds that of a hexagonal cell by an unexpectedly high value. From a topological point of view, the Bénard pattern obeys the Aboav-Weaire law at all times, even in the presence of threefold and fourfold vertices. The regimes above and below the transition are characterized by different topological correlations between neighboring cells. With the appearance of fourfold vertices, the topological correlation changes from negative to positive.