## Combining distance and area

What we are after is a measurement that reflects spaciousness and, importantly, when more things are built, reflects loss of spaciousness. As discussed on the previous page, area alone fails to reflect spaciousness; we need a measurement that includes both area and distance.

The first publication describing an effective way to measure space appeared in the journal *Science* in 2007 (see reference [1]). This study measured average distance to the nearest road and then multiplied the average distance by the area in question (a county or a state, for example). Why just distance to road, and not distance to any human-built feature? Most construction requires road access and most human-built features are, therefore, quite close to roads. This means that for a large area study, roads are an acceptable surrogate for all human constructions. Furthermore, digital road maps were readily available at the time of that publication, while maps that show all constructions are just now (2017) becoming common.

### Distance from what?

For large area studies^{1}, distance from roads works well because other human features are generally close to roads.The *foil* is the set of features to which shortest distance is measured. These features form the boundary of the uninterrupted space. For calculations of space in smaller areas, other features can be and generally should be included. Whatever features are included—we refer to these as the *foils* against which space is measured—need to be stated so that there is a clear understanding of what a particular space measurement means. A foil can be anything; for example, to calculate the space of a lake, the foil would be all land (in effect, the lake’s shoreline) and this might include interior land (islands). In other areas—on a prairie, for example—it might be important to include both fences and roads in the foil. In a coastal area, it might be appropriate to include roads and the coastline.

*Click or tap image to enlarge.*

### An example

In the adjacent illustration, black lines represent linear *foils*, perhaps roads or fences, and blue arrows extend from selected points (the tails of the arrows) to the nearest point on a foil. The length of each arrow is a measure of the minimum freedom of movement starting at its origin. For space calculations, the arrow starting points would be evenly distributed across the entire interior of the space—either on a regular grid or densely random.

### Space calculation

The simplest way to measure the average distance is to lay a grid of points on a map and to calculate the distance to/from the nearest foil features at each point that is in the tract whose space is being analyzed. In practice this is done using a Geographic Information System (GIS) and digital maps .

## The space calculation formula

To calculate space in a tract—this may be a single land parcel, a city, a county, a state, or the entire United States—do the following calculations:

- Determine
the average of the distance to the nearest feature (DTF)^{2}of the points in the tract.- Determine the area of the tract.
- Multiply the average distance by the area.

### Units of space measurement

The calculation defined above has units of *distance* × *area*, which is *volume*. We recognize that it is, at first, Space is measured as volume!hard to get used to the idea that space is correctly measured as a volume—but the volume metric reflects both area and distance and begins to make intuitive sense after working with it for a while (see Understanding Space Volume). Furthermore, space can be visualized effectively as volume and then this definition of space becomes obvious.

### Space algebra

The space in two adjacent counties, say, is the sum of the spaces of each county; this additive nature of space measurements follows directly from our calculation formula^{3}. The space in a state is the sum of the space of its counties. Likewise, the space of the United States or Mexico is the sum of the spaces of its component states, or in Canada the spaces of its provinces.

The practicality of doing these space measurements and additions over large areas was demonstrated by estimating the roadless volume of the 48 conterminous states, which is 2.1 million cubic kilometers[1].

### Read more…

Play with it—Space in your home
By moving your furniture, you can see how space-as-volume works. |
What is Space?
What is important about distance, and how distance yields area. |

Understanding Space Volume
Illustrations and explanations of the concept. |
Measuring Space I
Explains why area-only measurements are ineffective. |

Measuring Space (Technical)
A section of the web site explains how to do space calculations in a Geographic Information System. |
The Ecological Meaning of Space Volume
How do we interpret space, when measured as a volume, ecologically? |

__Footnotes:__

^{1}Reference [1] considered the entire area of the 48 conterminous states in the United States.

^{2}Here

*nearest feature*means the nearest feature in the

*foil*against which space is being measured. This may be roads only, or perhaps all permanent human-built features. The features that are included in the

*foil*need to be stated so that the user of the measurement knows exactly what it means.

^{3}For details, see the section of the web site starting with Measuring Space (Technical).