The natural ventilation of a room, both with a heated floor and connected to a cold exterior through two openings, is investigated by combining quantitative models with analogue laboratory experiments. The heated floor generates an areal source of buoyancy while the openings allow displacement ventilation to operate. When combined, these produce a steady state in which the air in the room is well-mixed, and the heat provided by the floor equals the heat lost by displacement. We develop a quantitative model describing this process, in which the advective heat transfer through the openings is balanced with the heat flux supplied at the floor. This model is successfully tested with observations from small-scale analogue laboratory experiments. We compare our results with the steady-state flow associated with a point source of buoyancy: for a given applied heat flux, an areal source produces heated air of lower temperature but a greater volume flux of air circulates through the room. We generalize the model to account for the effects of (i) a cooled roof as well as a heated floor, and (ii) an external wind or temperature gradient. In the former case, the direction of the flow through the openings depends on the temperature of the exterior air relative to an averaged roof and floor temperature. In the latter case, the flow is either buoyancy dominated or wind dominated depending on the strength of the pressure associated with the wind. Furthermore, there is an intermediate multiple-solution regime in which either flow regime may develop.