Why are rough inner city areas in the USA called "the projects"?

In the United States and Canada, public housing is usually a block of purpose-built housing operated by a government agency, often simply referred to as "projects."

https://en.wikipedia.org/wiki/Public_housing#United_States_and_Canada

Purpose-built eh? What a novel idea. Presumably thre are a lot of buildings built for no purpose whatever in the USA and Canada?

With respect to the 'rough' part of the question, the rough is more often the edifices themselves and not the people in them. Many building projects aimed at constructing affordable housing are just that; a projected solution to urban areas that have become dilapidated, for example: industrial areas, recovered bay waters in some cities, and many a dated hotel or apartment building. Good example in Seattle follows:
http://www.seattlehousing.org/landlords/project/