I'll also assume relatively flat and uniform terrain, and at most fair-weather cumulus clouds. Call the boundary layer height zi. The surface layer is heated by contact with the surface. So to first order, thermals are columns of warm and therefore buoyant air that rise from the surface layer to the inversion. The thermals themselves are somewhat less than half that, say 500-1000 m in diameter.
Most thermals span the boundary layer vertically.
There is, of course, a distribution of sizes. Between thermals are broad areas of sink. The sink is weaker than the lift because it covers a larger area. The opposite is true at the top of the boundary layer, but we rarely fly that high. There are, as always, complications.