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Certainly, there are Census definitions, but they, as do most measures, have faults. Why not population density? OK, that will help in agricultural areas, but how then to define a rural city/village? and what about areas on the rural fringe? A tough question. Still, I know it (rural) when I see it. How about "Where the cows start."
'Rural' can be a state of mind rather than a measurable application. It is quite possible to have a 'rural' enclave in the midst of a metro area (aka sprawl). Every area would have a different standard. In my neck of the woods, rural is in the zoning code: 80 acre parcels. Another is when you smell the cattle manure, you are rural. Perhaps 5 acres is more in line with your state. And have you defined 'urban center', a phrase used in your post? For us, that would be the central city of 25K pop.
Talk it over with your peers and especially non-planners. From those generalities, a definition may start to form. I think it may be the market economy rather than distance that becomes your criterion. Percentage of farm v. non-farm employment perhaps? But that wouldn't cover the second home hobby farms. Basic economic generators are agri-business? Just a couple of ideas.
Can there be one definition of rural? I think not. But keep in mind that Rural can also de defined by level of service. Sewer and water vs, private systems; mail delivery by carrier or rural route box; I wouldnt always trust one catch -all definition. Our zoning code(circa 1964) defines Rural Residential as 60,000 sqaure foot parcels! To me, that is not necessarily rural... You get the idea.
I think the missing question here is "why?" or for what purpose are you defining "rural." I normally say that rural is a place where the vast majority of folks actually make their living from the land they live on. This eliminates many many places that are called rural, and especially a lot of places that like to think of themselves as rural but will never properly address their land use issues until they admit that they are suburban or exurban. I counsel against using this word at all if it will contribute to a continuing misperception of the community's actual role in the world.
A few ways to define rural . . .these are some sample ideas.
Census - outside of a SMSA and outside of an incorporated political subdivision at the sub-county level.
Population - density of one dwelling unit per 5 acres or less as an average. Town size is less than 50,000 and population centers are separated by at least 30 miles
Economics - dominant labor category is agriculture and food related processing; livestock grazing including feedlots, open range livestock grazing
Utility providers are electrical coops, diesel, or wind v. line from utility, tank propone v. gas line, well v. city water, septic v. sewer
Transportation - two lane state or county highway, and no commercial development at Interstate Highway interchanges (excluding convenience store / gas) Few options for scheduled air transportation or charter air and private piloted aircraft only. Rail service, if any, is freight only
Land use - dominant form is agriculture, open range, or federal public lands including alpine / mountain and wilderness
In my consulting work, we often wax poetic about preserving "rural" characteristics, heretiage, etc. But pining down exactly what we mean can be difficult (as is pining down *anything* a consultant says).
Here's my wise-ass working definition of "rural character" for consideration.
"Tractor Day" at the high school
A day off from school for the start of deer season
Being able to pee off your back porch and not worry about what the neighbors think
The barbeque you eat Sunday afternoon was your neighbor's hog Friday morning
Knowing the difference between sweet corn and feed corn
Your dentist is also a large-animal vetranarian
I'm all for farmland preservation, at least until I want to sell my farm out for development.
When i grew up, I lived in a county of 130,000 people, it had 5 population centers of 10,000 plus and 2 of 20,000 plus. All the cities were within a 20 minute drive. It was a rather unique situation where there was no regional 'center' the area. it was nearly evenly spread between the cities. The whole area is also a good hour from any city over 50,000.
The county had a lot of people in it. All the cities were small, compact and rather dense towns that would qualify as urban density. But the region is considered rural (by the census) because it is outside of a cmsa/msa.
Rural can be defined on levels. What may be rural to Chicagoans may be diffrent from what someone from Po-dunk, Indiana thinks is rural.