Appendix A: Glossary

Central Place – The Census Bureau delineates an urbanized area (UA) as one or more “central places” and the “urban fringe” (the adjacent densely settled surrounding territory) that together contain a minimum of 50,000 residents.  A central place functions as the dominant center of each UA.  The identification of a UA central place permits the comparison of this dominant center with the remaining territory in the UA.  A central place generally is the most densely populated and oldest city in a metropolitan area.

Density – Shorthand for population density, or the number of residents per unit area, usually measured in number of residents per acre or square mile. Density is the mathematical inverse or opposite of land consumption per person (per capita).  For example, a density of five persons or residents per acre equals 3,200 per square mile. This in turn equals a per capita land consumption of 0.2 acre per person.

Developed Land – As defined by the U.S. Department of Agriculture’s Natural Resources Conservation Service in its National Resources Inventories (NRIs), issued every five years since 1982, built-up or paved land that is at least one-quarter acre in area. Developed land can include built-up areas outside of urbanized areas, towns, or cities.  The NRI Developed Land category includes: (a) large tracts of urban and built-up land; (b) small tracts of built-up land less than 10 acres in size; and (c) land outside of these built-up areas that is in a rural transportation corridor (roads, interstates, railroads, and associated rights-of-way).

Foreign Born – Describing a person born in a country other than the United States. Excludes those born abroad to American parents.  Can be used as a noun or an adjective.

High-Density – A large number of residents per unit area, usually measured in terms of residents per acre or square mile. While there is no one precise, agreed-upon criterion or threshold of high-density residential development, a density of approximately 5,000 per square mile would be considered relatively high-density.

Hop –  a connection from one urban area core to other qualifying urban territory along a road connection of half a mile (0.5 mile) or less in length; multiple hops may be made along any given road corridor.  This criterion recognizes that alternating patterns of residential development and non-residential development are a typical feature of urban landscapes.

Immigration – Permanent movement (i.e., settlement) of a foreign-born person to the

United States either with permission from U.S. authorities (legal immigration) or without such permission (illegal immigration).

Immigrant Fertility – Fertility of foreign-born immigrants to the United States, usually expressed in terms of the Total Fertility Rate (TFR) of women, which is the average total number of children born to women of a defined group during the course of their reproductive years.

Jump – a connection from one urban area core to other qualifying urban territory along a road connection between 0.5 mile and 2.5 miles in length; only one jump may be made along any given road connection.

Low-Density – Relatively low population density, or low number of residents per unit area (acre or square mile). Urban / suburban densities of 1,000-2,000 per square mile would be considered low-density, though still enough to qualify as urban.

Migration – Permanent movement or resettlement of an individual or group from one place to another.  Internal or domestic migration is from one city, state, or region to another within the United States.  International or foreign migration is when foreign-born individuals or groups move permanently, with or without permission (legally or illegally), into the United States from a foreign country.  In-migration refers to migration into the country from a foreign location or into a state from another state.  Emigration, or out-migration is the opposite.  

Native Born – A person born in the United States.

Natural Habitat – That portion of rural or undeveloped land that consists of upland and bottomland forests, woodlands, savanna, scrub-shrub, natural grasslands or prairie, wetlands (marshes, swamps, bogs), ponds, watercourses, deserts, alpine meadow and tundra.  Natural habitats support wildlife and provide other ecosystem services.  They may be in public or private ownership. 

New Urbanism – A movement that sees urban centers as potentially vibrant communities that can mix and harmonize residential and commercial uses in clever and innovative ways to make cities satisfying and safe places to live and work.  New urbanism supports such concepts as higher density in urban cores, mixed uses, mass transit, close proximity of dwellings to workplace, walkable communities, bicycle lanes, community gardens, and others.  New urbanism sees relentless sprawl in America as one consequence of the abandonment of our central cities.

Open Space – Land lacking significant built structures or pavement.  Includes rural and undeveloped lands and natural habitat outside of urban boundaries; also includes larger natural areas, parks and green space within urban areas, such as golf courses and extensive lawns or gardens.  Yards or wooded lots on quarter-acre lots in residential areas would not qualify as open space.  

Overall Sprawl  – See “sprawl” below.  Overall sprawl is the sum of Per Capita Sprawl and Population-driven sprawl [the total amount of open space converted to development over a period of time].  

Per Capita Land Consumption – Average amount of land used by each resident of an urbanized area or developed area.  Includes not just residential land but all developed land used by urban residents, including commercial, institutional, small park, transportation (e.g., streets, roads, railroads, freeways, parking lots), and industrial land uses.

Per Capita Sprawl – Sprawl that is driven by increase in per capita land consumption, that is, land consumption per resident, of an urbanized area, developed area, city or town; Per Capita Sprawl is measured in terms the increase in acres or square miles of developed or urbanized acres of land per person.  Per Capita Sprawl and population-driven sprawl add up to 100 percent of Overall Sprawl.

Population-driven Sprawl – Sprawl that is driven by increase in the population of an urbanized or developed area.  Population-driven and Per Capita Sprawl add up to 100 percent.

Population Growth – Increase in the number of residents of a given area, such as a town, city, urbanized area, state, or country over time. Population growth is equal to the total births of native-born residents minus the total deaths of native-born residents minus the emigration of native-born residents PLUS total immigration of the foreign born plus births to the foreign born minus deaths of the foreign born minus emigration of the foreign born (i.e., return to the country of their birth or a third country).  In recent decades, annual population growth in the United States as a whole has been running about 2.5 million to 3 million per year on average, or roughly 30 million per decade.  In the 2010 to 2020 decade, U.S. population growth was approximately 23 million, a notable decline from the previous two decades, but still far from “population stabilization” (zero population growth). 

Rural Land – Undeveloped lands outside of urban areas, including farmland, pastureland, rangeland, and natural or semi-natural habitats, like forests, woodlands, wetlands, grasslands or prairie, and deserts.  Rural lands may be flat or mountainous, and publicly or privately owned.

Smart Growth – The use of a variety of land-use, planning, statutory, regulatory, taxing, and other tools by federal and state governments and local jurisdictions (municipalities) to reduce haphazard, low-density, and poorly planned development in a given region.

Smart Growth Movement – A loose, eclectic coalition of environmentalists, local growth-control activists, New Urbanists, municipal and regional planners, think-tanks, the federal government and many state governments, and even some home-builders and developers united by their interest in slowing the rate of sprawl, and making existing communities more sustainable and livable.

Sprawl – As defined in this study, the increase in the physical area of a town or city over time – outward expansion – as undeveloped or rural land at its periphery is permanently converted to developed or urbanized land as population and/or per capita land consumption grow.  More specifically, in this study, sprawl is 1) the increase in the area of the Census Bureau’s Urbanized Areas, as delineated every 10 years in the decadal censuses, and/or 2) the increase in the area of a state’s area of Developed Land, as determined by the Natural Resources Conservation Service. 

Suburbs – Residential or commercial zones on the outskirts of a central city or town; generally corresponds to “urban fringe.”  Tend to have a lower population density than the central place or urban core, though not always, as when downtown districts are dominated by office, institutional, and commercial zones.  

Urban Core – Used in this report as another description for “central location” as defined by the Census Bureau. The urban core is the entire city that anchors a metropolitan area, and usually is at its center. It generally is the oldest, most densely populated and most built-up portion of an urbanized area.

Urban Fringe – Built-up areas near the edge of an urbanized area, generally with lower population density than the urban core; generally corresponds to the inner and outer suburbs of a town or city.

Urban Sprawl – See “sprawl.”

Urbanized Area – As defined by the U.S. Census Bureau, an area of contiguous census blocks or block groups with a population of at least 50,000 and an average population density of at least 1,000 residents per square mile.


Appendix B: Calculating Per Capita Land Consumption

Calculating Per Capita Land Consumption

The per person land consumption in each state or Urbanized Area can be expressed as:

(1) a = A / P


a = area of developed or urbanized land area for the average resident

A = Area of total developed or urbanized land in a state

P = Population of that state


For example, in 2017 the 48 coterminous states and Hawaii had a combined population of 323,550,933 residents and approximately 115,726,400 developed acres. Thus, per capita developed land use for all purposes was around 0.358 acre (slightly more than a third of an acre) per resident.


The land used per person is the total developed or urbanized land area divided by the total number of people. This is the inverse of population density, which is the number of people per unit area of land. When per capita land consumption goes up, density goes down; when per capita land consumption goes down, density goes up.


The developed land area of any given state can be expressed as:


(2) A = P x a


This can be stated as: the total developed area in square miles (or acres) of a county or state can be simply expressed or “factored” into the product of the Population of the county or state (viz., P) multiplied by the per capita urban land consumption (viz., a). This second equation (2) is the basis for attributing or apportioning the shares of sprawl (viz. growth in A) back onto two contributing factors, the growth in P and the growth in a.




Appendix C: Apportioning Shares of Overall Sprawl Between
Population Growth and Per Capita Sprawl

Apportioning Shares of Overall Sprawl Between 

Population Growth and Per Capita Sprawl

A methodology for quantifying the respective contributions of population growth and changes in per capita consumption of any type of resource use was outlined in a 1991 paper by physicist John Holdren (“Population and the Energy Problem.Population and Environment, Vol. 12, No. 3, Spring 1991).   Although Dr. Holdren’s 1991 paper dealt specifically with the role of population growth in propelling the increase in U.S. energy consumption, the same methodology can also be applied to many types of population and resource consumption analyses. 

In the case of sprawl, the resource under consideration is rural land, namely the expansion over time in the total acreage of rural land urbanized or converted into developed land and subsequently used for urban purposes, such as for housing, commerce, retail, office space, education, light and heavy industry, transportation, and so forth.   

As stated in Appendix B, the total land area developed in a city (urbanized area), county, or state can be expressed as:

(1) A = P x a


A = Area of total are (in acres or square miles) of development in city or state

P = Population of that city or state

a = area of city or state used by the average resident (per capita land use) 

Following the logic in Holdren’s paper, if over a period of time Δt (e.g., a year or a decade), the population grows by an increment ΔP and the per capita land use changes by Δa, the total urbanized land area grows by ΔA, expressed as:

(2) A + ΔA = (P + ΔP) x (a + Δa)

Subtracting eqn. (1) from eqn. (2) and dividing through by A to compute the relative change (i.e., ΔA/A) in urbanized land area over time interval Δt yields:

(3) ΔA/A = ΔP/P + Δa/a + (ΔP/P) x (Δa/a)

Now equation (3) is quite general and makes no assumption about the growth model or time interval.  On a year-to-year basis, the percentage increments in P and a are small

(i.e., single digit percentages), so the second order term in equation (3) can be ignored.

Hence following the Holdren paradigm, eqn. (3) states that the percentage growth in an urbanized land area or developed area of a country (viz., 100 percent x ΔA/A) is the sum of the percentage growth in the population ( 100 percent x ΔP/P) plus the percentage growth in the per capita land use (100 percent x Δa/a). Stated in words, equation (3) becomes:

(4) Overall percentage land area growth = Overall percentage population   growth + Overall percentage per capita growth

In essence, this apportioning methodology quantifies population growth’s share of total land consumption (sprawl) by finding the ratio of the overall percentage change in population over a period of time to the overall percentage change in land area consumed for the same period. This can be expressed as:

   (Overall percentage population growth)

(5) Population share of growth =   (Overall percentage land area growth)

The same form applies for per capita land use:

  (Overall % per capita land use growth)

(6) Per capita land use share of growth =   (Overall % land area growth)

The above two equations follow the relationship based on Prof. Holdren’s equation (5) in his 1991 paper.  A common growth model follows the form (say for population):

(7) P(t) = P0 (1 + gp)t

Where P(t) is population at time t, P0 is the initial population and gp the growth rate over the interval.  Solving for gp the growth rate yields:

(8) ln (1 + gp) = (1/t) ln (P(t)/P0)

Since ln (1 + x) approximately equals x for small values of x, equation (8) can be written as:

(9) gp = (1/t) ln (P(t)/P0)

The same form of derivation of growth rates can be written for land area (A) and per capita land use (a)

(10) gA = (1/t) ln (A(t)/A0)

(11) ga = (1/t) ln (a(t)/a0)

These three equations for the growth rates allow the result of equation (4) to be restated as:

(12) gP + ga = gA

Substituting the formulae (equations 9 through 11) for the growth rates and relating the initial and final values of the variables P, a and A over the period of interest into equation (12), the actual calculational relationship becomes:

(13) ln (final population / initial population) + ln (final per capita land area / initial per capita land area) = ln (final total land area / initial total land area)

In other words, the natural logarithm (ln) of the ratio of the final to initial population, plus the logarithm of the ratio of the final to initial per capita land area (i.e., land consumption per resident), equals the logarithm of the final to the initial total land area.

In the case of the United States (49 states) from 1982 to 2017, this formula would appear as:

(14) ln (230,580,652 residents / 323,550,993 residents) + ln (0.312 acre per resident / 0.358 acre per resident) = ln (71,847,500 acres / 115,726,400 acres)

Computing the ratios yields:

(15) ln (1.403) + ln (1.148) = ln (1.611) 

0.339 + 0.138 = 0.477

Then, applying equations (5) and (6), the percentage contributions of population growth and per capita land area growth are obtained by dividing (i.e., normalizing to 100 percent) each side by 0.54381:

(16) 0.339   +    0.138       = 0.477

0.477          0.477   0.477

Performing these divisions yields:

(17) 0.71 + 0.29 = 1.0

Thus, we note that in the case of the 49 states of the USA (all except Alaska) from 1982 to 2017, the share of sprawl due to population growth was 71 percent [100 percent x (0.339 / 0.477)], while declining density (i.e., an increase in land area per capita) accounted for 29 percent [100 percent x (0.138 / 0.477)].  Note that the sum of both percentages equals 100 percent.

In the case of Colorado,  we modify this gross state-wide percentage of sprawl related to population growth by using a county-by-county weighting approach.  This approach accounts for the sprawl that occurs in each county and applies a proportionately greater weight to those counties with greater amounts of sprawl.  In essence, sprawl in counties around Denver for example, should not be attributed to population growth in counties around Colorado Springs or Fort Collins.  

In this method, the amount of sprawl related to population growth in each county is summed for all of the counties in the state.  This sum or aggregate is then divided by the total amount of sprawl in the state.  Using this procedure, by way of example, 86 percent of the sprawl in Colorado between 1982 and 2017 was shown to be related to population growth, which the authors believe is a more accurate rendering of population growth’s role than 100 percent, which exaggerates population’s role, and implies that all sprawl in Colorado during that period was related to population growth, which was not the case.