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It seems like the common theme is about 2-3 sq miles for major cities. Center City in Philly, Downtown Atlanta, Downtown Minneapolis are all +/- 2.5 sq miles.
There are some exceptions, Boston has an exceptionally tiny "Downtown" only about .5 of a square mile. Downtown Cincinnati is also under 1 sq mile.
The area regarded as downtown Raleigh today currently spans approximately 754 acres, or 1.18 square miles. The geographic area that encompasses downtown constitutes less than one percent of Raleigh's incorporated area.
“Downtown” LA is 4.73 sq miles. San Diego, 2.3 excluding the Tidelands. Chicago's Loop is 1.58. Indianapolis’s Downtown “Mile Square” is 6.5. In many cases, it really seems to come down to how a city, over its history and development has defined “downtown.” Some are based on initial city charter plots, like Raleigh and Indianapolis, some are naturally hemmed in by topography like Boston, and others by man himself by rail or highway, like Chicago or LA.
In Europe it was often easy to define, as it was where the citizens (or their overlords) chose to wall themselves in.