To gain the maximum internal floor area in the most cost-effective manner, the ideal height and shape of a house would be a two-story rectangular or square structure with a simple gable or hip roof.

Here’s why this configuration is considered the most cost-effective for maximizing floor area:

1. Height: A two-story structure is generally more cost-effective than a single-story or taller multi-story building. The additional floor area gained from the second story is achieved with a relatively small increase in foundation, roof, and exterior wall costs compared to a single-story home of the same footprint.
2. Shape: As mentioned earlier, a rectangular or square shape is the most efficient in terms of maximizing interior floor area while minimizing exterior surface area, which reduces material and construction costs.
3. Roof style: A simple gable or hip roof is typically the most cost-effective roof design. These styles have straight lines and fewer complex angles or curves, which require less framing and roofing material compared to more intricate roof designs.

By combining a two-story height with a rectangular or square footprint and a simple gable or hip roof, you can effectively double the floor area while minimizing the additional costs associated with exterior walls, foundations, and roofing materials.

It’s important to note that this configuration assumes a relatively level building site and standard construction methods. Factors such as sloping terrain, local building codes, or specific design preferences may necessitate modifications or additions that could impact the overall cost-effectiveness.

Additionally, while this approach maximizes floor area, it may not be the most energy-efficient or aesthetically pleasing design. Trade-offs between cost, functionality, and personal preferences should be considered when determining the optimal height, shape, and design of a house.