GUIDE TO MAP SCALE

(note: This is a series of exerts from lots of various sites)

This guide has been prepared to help the users to understand the concept of scale in cartography.  Scale is defined as the ratio of the distance on a map to the corresponding distance on the surface the map represents.

I. THREE FORMS OF SCALE.  Scale is shown on maps in three ways:

A. Verbal Scale:

1 inch equals 16 miles

This example tells us that 1-inch on the map represents 16 miles on the surface of the Earth. This is the easiest scale to understand because it generally uses familiar units.

B. Graphic or Bar Scale:

______________________________________
_____16________0________16_________32_ miles

The Bar Scale is particularly important when enlarging or reducing maps by photocopy techniques because it changes with the map. If the Bar Scale is included in the photocopy, you will have an indication of the new scale.

C. Representative Fraction (RF) or Natural Scale:

1:1,000,000 (this is the same as 1/1,000,000)

The RF says that 1 of any measurement on the map equals 1 million of the same measurement on the original surface; for the example above 1-foot equals 1 million feet or 1 cm. equals 1,000,000 cm. This is the form of scale commonly used in the Map Collection. A good quality map should have both the RF and Bar Scales.

II. LARGE AND SMALL SCALE: when we speak of large-scale maps we are saying the RF is large, i.e. the RF's denominator is small. 1:10,000 and 1:62,500 maps are large scale. Small-scale maps have a small RF. 1:500,000 and 1:1,000,000 maps are small scale.

III. HOW TO CONVERT FROM ONE FORM OF SCALE TO ANOTHER. Often when using cartographic materials it is useful to convert from one form of scale to another. If you have a good understanding of the concept of scale, the techniques are fairly simple. Here is an example of converting from Verbal Scale to RF. Remember; the RF has the same unit of measurement on both sides of the colon.

1 inch equals 10 miles

1 inch = 10 miles

1 inch = 10 miles x 12 inches/foot x 5280 feet/mile

1 inch = 10 x 63360 inches = 633,600 inches

1:633,600

To convert from RF to Verbal Scale you convert the fraction to familiar units of measurements; for example:

1:250,000

1 inch = 250,000 inches

1 inch = 250,000 inches [d] 12 inches/foot = 20,833.3 feet

1 inch = 20,833.3 feet [d] 5280 feet/mile = 4 miles or

1 inch = 250,000 [d] 63360 inches/mile = 4 miles

1 inch equals 4 miles

[Note:  [d] = divided by]

IV. SOME COMMON SCALES. Here is a list of RF scales commonly used in the Map Collection and their equivalent Verbal Scales.

1:24,000  - 1 in. = .379 mi.   1:250,000   - 1 in. = 4 mi.

1:62,500  - 1 in. = .986 mi.   1:500,000   - 1 in. = 7.891 mi.

1:100,000 - 1 in. = 1.578 mi.  1:1,000,000 - 1 in. = 15.783 mi.

V. HOW TO DETERMINE WHICH SCALE MAP WILL FIT ON A PARTICULAR SIZE OF PAPER. At times map users need to calculate the scale of a map that will fit on a particular size of paper. To do this you first need to determine the dimensions of the area you want on the map and then relate that to the dimensions of the new sheet.

For example you want a map of Arizona on a 8 1/2 x 11 inch piece of paper. To allow for 1/2-inch margins the new sheet will then be 7 1/2 x 10 inches. Since Arizona's north-south dimension, 395 miles, is slightly longer than its east-west dimension, 340 miles, we will place the longer north-south dimension along the longer 10-inch dimension of the paper. The next step is to compute the scales for both dimensions of the State. The smaller of the two scales will be the one we need.

North-south                                                  East-west

10 in = 395 mi                                              7.5 in = 340 mi

10 in = 395 mi x 63360 in/mi                       7.5 in = 340 mi x 63360 in/mi

10 in [d] 10 = 25027200 in [d] 10               7.5 in [d] 7.5 = 21542400 in [d] 7.5

1 in = 2502720 in                                        1 in = 2872320 in

1:2,502,720                                             1:2,872,320

[Note:  [d] = divided by]

We therefore need a map of Arizona at a scale of 1:2,872,320 or less to place it on an 8 1/2 x 11 inch sheet of paper.

VI. HOW TO FIND MAPS AT A PARTICULAR SCALE IN THE MAP COLLECTION. Maps cannot be located in the online catalog directly by scale. You need to look under the geographic area or under a thematic subject heading to see what maps are available. Map scales are given in the catalog in the RF form. A map series for a larger area may include the area you are interested in; so be sure to check for maps of larger areas such as countries or continents. For example, the entry for the 1:100,000 series topographic maps for Arizona may be found only under the heading "United States-maps, topographic". If you need assistance in locating a map at a particular scale please ask a staff member at the reference desk.

VII. TOPOGRAPHIC MAPS FOR ARIZONA. Here is a list, from the largest scale to the smallest, of the various series of topographic maps available for Arizona.

• 1:24,000. 7.5-Minute Series (Topographic). Reston, VA: U. S. Geological Survey, 1945 to present, app. 1971 sheets when complete. G4331s.C2 1882.U6
• 1:50,000. Arizona 1:50,000 [15 Minute Series (Topographic)]. Washington, DC: Army Map Service and Defense Mapping Agency, 1947 to present, currently 92 sheets, coverage incomplete. G4330s.50.U5
• 1:62,500. 15-Minute Series (Topographic). Reston: USGS, 1910 to 1968, 306 sheets, coverage incomplete. G4331s.C2 1882.U6
• 1:100,000. 30 x 60 Minute Series (Topographic). Reston: USGS, 1980 to present, 68 sheets when complete. G3700s.100.U5
• 1:125,000. 30-Minute Series (Topographic). Reston: USGS, 1901 to 1939, 23 sheets, only partial coverage. G4331s.C2 1882.U6
• 1:250,000. 1 x 2 Degree Series (Topographic). Reston: USGS, 1953 to present, 22 sheets, periodically revised. G4050s.250.U5
• 1:500,000. State of Arizona. Reston: USGS, 1981, 1 sheet. G4330. 1981.G4
• 1:1,000,000. World (North America) 1:1,000,000. Reston: USGS, 1952, 4 sheets, coverage incomplete. G3200s.1, 000.U51

What is Map Scale?

Map scale is the relationship between a unit of length on a map and the corresponding length on the ground. It's also an expression of how much the area represented has been reduced on the map. Map scale is important for understanding maps both in paper and computer form, so it will pay you to understand the types and uses of scales.

Types of Map Scales

We can relate map and ground with three different types of scale.

Verbal scale expresses in words a relationship between a map distance and a ground distance. Usually it is along the lines of:

One inch represents 16 miles.

Here it is implied that the one-inch is on the map, and that one-inch represents 16 miles on the ground. Verbal scales are commonly found on popular atlases and maps.

The second type of scale is a graphic scale, or bar scale. This shows directly on the map the corresponding ground distance. For example:

Bar scales is probably the most common kind of scale found on maps, perhaps because their graphical nature makes them easily understood? Another great thing about bar scales is that they remain correct if the map is reduced or enlarged photographically. This is not true of the other two types of scales.

The third type of scale is a representative fraction, or ratio scale. Compared to the first two, it is the most abstract, but also the most versatile. A representative fraction, or RF, shows the relationship between one of any unit on the map and one of the same units on the ground. RFs may be shown as an actual fraction, for example 1/24,000, but are usually written with a colon, as in 1:24,000. In this example, one unit of any length (one mm, one cm, one inch, one foot, etc.) on the map represents 24,000 of those same units on the ground (24,000 mm, 24,000 cm, 24,000", 24,000', etc.). The RF is versatile because you are not tied to any specific units. You may work in any unit you choose, metric, English, or other.

The RF is a called a fraction because it is just that--a fraction that shows how much the real world is reduced to fit on the map. A good comparison is often made with scale models of automobiles or aircraft. A 1/32-model of an auto is 1/32nd as large as the actual auto. In the same way, a 1:100,000-scale map is 1/100,000th as large as the ground area shown on the map.

Many people get confused by the abstract nature of RF scales. They ask, "one what is equal to 24,000 of what?" The key is that you can put in any unit you want instead of "what" in the question. That's what makes it the most flexible type of scale.

RF scales are common among geographers as shorthand for types of maps. For instance, the common local-area quadrangle maps produced by the US Geological Survey, which each cover 7.5 minutes of latitude and longitude, are printed at 1:24,000. Hence these are often referred to as 1:24,000 maps, or even "24K quads."   Many maps include two or even all three types of scales. USGS topographic maps have both bar scales and RFs.

##### Small Scale Vs. Large Scale

A related idea is that of small scale versus large scale. Geographers use these terms differently than many people. A large-scale map is where the RF is relatively large. A 1:1200 map is therefore larger scale than a 1:10,000,000 maps. The 1:10,000,000 maps would usually be called a small-scale map. This is true even though the 1:10,000,000 maps would show a much larger area than the 1:1200 maps. Here is a rule of thumb for size of scale by RF:

Large Scale: 1:25,000 or larger

Medium Scale: 1:1,000,000 to 1:25,000

Small Scale: 1:1,000,000 or smaller

Of course, what is small or large scale is relative. I noticed a surveying text (Brinker & Wolf 1984) that classed anything smaller than 1:12,000 as small scale -- surveyors rarely work with anything smaller than this.

The large/small scale terminology can become confusing when talking about large versus small areas. If you are talking about a phenomenon that occurs across a large region, it is tempting to say it's a large-scale phenomenon (e.g., "the forest blight is a large-scale disease"). But since the map that would show this would be small-scale, it is better to use a different term to avoid confusion. My favorite is "broad-scale."

##### Determining Distance and Area from Map & Scale

Map scale isn't much use in and of itself. We can use a map's scale to determine distances and areas on the map. Compared to converting between scale types, calculating distance is simple. Area calculations are trickier, since we have to square the numbers.

##### Finding distance from map and scale

As an example, suppose we have a map with a scale of 1:50,000. We measure the distance along a property boundary as 1.7 cm. What is the length in the real world?

To find ground distance, we must use the map scale to convert map distance to ground distance. Notice that again we inverted the RF scale, so the units will cancel properly. Once we multiply by the scale, we need to convert the ground distance to units suitable for ground measurement--in this case, from centimeters to kilometers.

We can also calculate distance from verbal and graphic scales. With verbal scales, we use the same procedure as above with the RF. The only difference is that we have to use the units given in the verbal scale (e.g., 1 inch to 17 miles). We'd probably want to measure our map distance in the same units (in this case, inches) to make our conversion easy.

Graphic scales are probably the scales most frequently used by laypersons. You can mark off a distance on the map and compare it directly to the bar scale. You need not know how many inches or centimeters the map distance is. The main drawback of bar scales is that they are usually short compared to the map itself, and hence measuring longer distances is difficult.

##### Finding area measurement from map and scale

Area must be expressed in areal units, which are usually distance units squared -- cm2, mi2, and so on. We must therefore used squared conversion factors when finding area from map measurements.

For example, suppose we measure a rectangular piece of property that is 3 cm by 4 cm on a map. The map is at a scale of 1:24,000. What is the area of the parcel?

The area of the parcel on the map is 3 cm * 4 cm = 12 cm2.

The area is 691,200 square meters on the ground. Since this is a large number, we might want to translate to other units. There are 10,000 square meters per hectare, so the area is 69 hectares (ha) (a hectare is about 2.5 acres). Or, there are (1,000)2 = 1,000,000 square meters per square kilometer, so the area is also 0.69 km2.

Notice that by writing the units as part of the problem, and squaring them along with the numbers, our units cancel properly and we end up with a sensible answer.  There is another way to tackle area problems if you have distance dimensions like 3 x 4 cm to start out the problem. You can convert the distance dimensions to real-world distances first, and then multiply them to find the area. This makes the problem longer but perhaps simpler.

##### Converting Between Scale Types

If you are given one type of scale, you may need to derive or construct any of the other two. This takes some practice. Some examples are given below.

A vital step in doing any kind of conversion that involves differing units is to include the units in the problem itself. You can then cancel the units by multiplying or dividing. This way you avoid becoming confused about which conversion factors to use and how to use them.

##### Verbal Scale to RF

The key here is to write the verbal scale as a fraction, and then convert so that both numerator and denominator have the same units, and the numerator has a 1.  Convert verbal scale of "1 inch represents 18 miles" to RF.

or 1:1,140,000.

Notice that the resulting fraction is rounded so that the RF does not imply more accuracy than the original precision warranted.  Convert verbal scale of "15 cm to 1 km" to RF.

, or 1:6700.

In many conversions you can save steps if you remember additional equivalencies. For example, in (a) above, we could have used the fact that 1 mile = 63,360 inches to skip a step.

##### Verbal Scale to Graphic Scale

Usually this is a relatively easy task if the map gives us reasonable units in the verbal scale. We can use the verbal scale like a fraction to transform the ground distance to map distance.

Convert verbal scale of "1 cm to 14 km" to a graphic scale.  One centimeter is a relatively small distance, so we probably don't want our bar scale to have major divisions much smaller than this. A centimeter represents 14 km, so a division of 10 km is probably fine. Therefore we want to find how many centimeters represent 10 km.

In other words, we can represent our 10 km increment on the bar scale by measuring off 0.71 cm on the map. We'd draw the first tick at 0.71 cm, the second at 1.42 cm, and so on (note: the scale below will vary in size depending on your screen resolution and size):

RF to Graphic Scale

This adds an extra step to the example above. We can find the map-distance equivalent of a ground distance, but we also need to be careful about choosing which ground distance we want to portray on the map. Perhaps it's easiest to choose a smaller ground distance that you can then multiply to get a reasonable bar scale.

Convert an RF of 1:250,000 to a graphic scale.  If we aren't sure what increments a bar scale would have for this scale, we could start out, say, with finding the map equivalent of 1 mile:

This might work fine, with one mile marked off on the map every 0.25-inch; or, we may want finer or broader increments, which we can find by dividing or multiplying the .25" as needed. We would draw the scale the same way as in the previous example.

##### RF to Verbal Scale

Again we have to choose appropriate units to convert into. Most verbal scales are either "one inch represents ____ miles," or "one centimeter represents ___ kilometers." These are relatively easy to do, since it means only that we convert the denominator of our RF to the larger units.  Convert from RF of 1:25,000 to a verbal scale, in metric.

Therefore, one centimeter on this map represents 1/4 of a kilometer on the ground (or, 10 centimeters represents 2.5 kilometers, etc.).

##### Graphic Scale to RF

Here we must take a measurement from the bar scale to determine the map distance that corresponds to a ground distance.  Find the RF scale for the following graphic scale:

By measuring with a ruler (your results may differ depending on screen resolution and size!), we find that 10 kilometers measures 2.4 cm. We can use this relationship to find the RF for the bar scale:

, Or 1:420,000.

##### Determining Scale from a Map or Photo

Some maps may come with no scale at all. Aerial photographs almost never do (unless one was painted on the ground before the photo was taken!). How can you derive a scale for use with the map or photo?

##### By measuring object of known size on map or photo

Actually the procedure is very similar to the last example above. But instead of measuring along a bar scale, you must measure the length of an object on the map or photo whose actual length you know. This might be a football field, a city block, or the whole Equator (if it's a world map). Often you can identify 1-mile-square sections in the US (in areas under the Public Land Survey System, which is most of the US other than the East Coast, Texas and parts of coastal California). You may even need to go out to the location mapped or pictured and measure the distance between two identifiable objects. Once you have the two distances, you can find the scale as in the above example for graphic-to-RF scale.

For another example, suppose you have a map where the distance between two section-line roads is 3.5 inches on the map. We can usually assume this distance is one mile on the ground (there are exceptions). The RF scale is then

, Or 1:18,000.

One caveat (exception) for air photos is that this method assumes the two locations are at the same elevation--or that the terrain is flat. If you are using air photos, the terrain may not be flat. If there are hills, even moderate ones, the calculations can be thrown off due to relief displacement. See references on aerial photography and photogrammetry for more information.

##### By comparing with another map or photo of known scale

Another way to calculate scale on an unknown map or photo is to compare it to a map with a known scale. For example, suppose you have an air photo where the distance between two hills is 7.2 centimeters. You have a map of the same area at 1:24,000, and on the map the distance between the hills is 2.4 centimeters. The answer involves a little algebra. Since the ground distance is the same on both photo and map, we can create an expression for this ground distance for both, and then put them on either side of an equation. The ground distance can be found by multiplying the map/photo distance by the scale (in this case, by the inverse of the scale--notice how this makes the units cancel correctly). We need to find, for the photo, how many ground units are represented by one unit on the photo, so we use an x for this unknown quantity and solve for it:

,

And 7.2 * x cm ground = 2.4 * 24,000 cm ground; we can cancel the units on each side and divide by 7.2: x = (2.4 * 24,000)/7.2 = 8000.  In other words, the RF scale for the photo is 1:8,000.

##### Scale, Accuracy, and Resolution in a GIS

Note: This document was created in 1995 as a general introduction. A section on precision was added in 1998. See Error, Accuracy, and Precision at The Geographer's Craft Project for a much more comprehensive treatment.

Because GIS data is stored in a very different way than paper map data, the relationships between map scale, data accuracy, resolution, and density are very different between GIS and paper maps.

##### Map scale

Map scale specifies the amount of reduction between the real world and its graphic representation (usually a paper map). It is usually expressed as a ratio (e.g. 1:20,000), or equivalence (e.g. 1 mm = 20 m). Since a paper map is always the same size, its scale is fixed when it is printed, and cannot change.

However, a map in a GIS can be shrunk or enlarged at will on the screen or on paper. You can zoom in until the screen displays a square meter or less, or zoom out until the screen displays all of BC. This means that geographic data in a GIS doesn't really have a 'map scale'.

##### Display scale

The display scale of a map is the scale at which it 'looks right'. Because a paper map is created at certain scale, its 'map scale' and 'display scale' are the same. The display scale influences two things about a map: The amount of detail. The map must not be overwhelmed with detail, and become too crowded. The size and placement of text and symbols. These must be sized to be readable at the display scale, and placed so that they do not overlap each other.

If you put a 1:20,000 scale paper map on a reducing photocopier, you can make it into a 1:100,000 map (i.e. reduce it by a factor of 5). However, probably areas of detail will be merged into big black blobs, and most of the text on the map will be too small to read.

A GIS map's annotation (i.e. text and symbols) must be designed with a display scale, just like a paper map. There is a range of scale in which it will 'look right', even though it is possible to display it at other scales with the GIS software.

• Data accuracy and uncertainty
• Data accuracy is a statement of how closely a bit of data represents the real world. It applies to geographical information in all these ways:
• What features have been omitted?
• What non-existent features are represented?
• How correct is their classification?
• How current is the data?
• How far away is a map feature from its actual location in the world?
• This last, or 'locational' accuracy is of interest here, and is generally stated in terms of uncertainty. Locational accuracy includes:

Absolute accuracy: How close is the location on the map or data representation to its real location on the earth? For example, '95% of the well locations are within 50 meters of their surveyed locations'.

Relative accuracy: How similar is a shape on the map or data representation to the shape of the object on the earth. For example, 'cutblock boundaries do not vary by more than 10 meters from their actual shape'. These are separated because a map object may have a very accurate shape, but not be registered (located) correctly.

A rigorous statement of accuracy will include statistical measures of uncertainty and variation, as well as how and when the information was collected. Spatial data accuracy is independent of map scale and display scale, and should be stated in ground measurement units.

##### Data precision

Data precision is the smallest difference between adjacent positions that can be recorded and stored. Most GIS store locations in ground units (e.g. UTM coordinates, or Longitude/Latitude) with a precision of a meter, centimeter or less.

##### Data resolution

Resolution is the degree to which closely related entities can be discriminated. Since a paper map is always the same size, its data resolution is tied to its scale. Resolution also limits the minimum size of feature that can be stored. Generally, a line cannot be drawn much narrower than about 1/2 a millimeter. Therefore, on a 1:20,000 scale paper map, the minimum distance which can be represented (resolution) is about 10 meters. On a 1:250,000-scale paper map, the resolution is 125 meters.

Usually, it is desirable to specify the resolution of a dataset as a minimum feature size. For example, 'no lakes of less than 5 hectares surface area should be captured'. In a GIS, this is the most important reason for having the same data represented at different 'scales'.

##### Raster data resolution

Raster data is stored as (usually square) pixels, which form a grid or mesh over an area of the earth. The size of these pixels determines the resolution of the raster, because it is impossible to store anything, which falls 'between' the pixels. A GIS allows raster pixels to be any size, although they should not be smaller than the uncertainty of the data. If raster coverage is derived from vector line work, its pixels should not be smaller than the uncertainty in the line work. If it comes from an air-photo or satellite image, its pixels should not be smaller than the resolution of the camera that recorded it.

##### Data density

Data density is a measure of how many features per area are stored, and may imply a minimum feature size. Greater density implies more features in a given area, and therefore the features may be smaller.  The density of paper map's data is limited by its scale (and therefore its resolution). Areas (polygons) cannot be shown if they are smaller than the lines, which draw them. For example, a polygon less than 250 meters wide cannot be drawn on a 1:250,000-scale map. This minimum size also limits the number of polygons that can be represented in a given area of a paper map.

##### Data detail

Data detail is a measure of how much information is stored for each feature. A GIS stores lines (e.g., a lake shoreline) as a sequence of point locations, and draws it with the edges that join them. There is no limit to how many points can be stored, or how close together they may be. The amount of detail on line features should be limited just like data density. It does not make sense to store points at intervals, which are shorter than the accuracy of their locations.

##### GIS analysis

In a GIS, analysis is done at the precision of the data, not at any display scale. For example, the area of a habitat polygon is calculated to the nearest square centimeter. The GIS will carry much more precision through its calculations than are justified by the data's accuracy. The results of these calculations should be rounded to a value appropriate to the uncertainty of the data for reporting.  Some operations may result in features, which are smaller than the data uncertainty. For example, overlaying rivers and forest polygons may create 'slivers' along the riverbanks, which are 10 meters wide, when the uncertainty of the data is 20 meters. These slivers should be ignored, or included with their neighbors before the results of the overlay are used for further analysis.

Separation of data and annotation

In a GIS, it is common to display the same data (e.g. wildlife management unit boundaries) at several different scales for different purposes. It is also possible to create symbols and text that 'look right' at several different scales, and store them apart from the data they label. For example, management unit boundaries could be stored in one provincial coverage, and annotation layers could be developed for labeling them at display scales of 1:20,000, 1:250,000, and 1:2,000,000. If done carefully, this avoids duplication of the same data for display at different scales.

Generalization

In a GIS, it is possible to create a new coverage by reducing the amount of detail in existing coverage. This 'generalizing' may or may not reduce the number of objects in the coverage. For example, a detailed forest cover map may be generalized by combining polygons with similar characteristics. This reduces the number of objects in the coverage.

Conversely, a detailed ecosystem classification map may be generalized by reducing the amount of detail in the boundaries between regions, without reducing the number of regions. Generalizing a raster image usually reduces both the number of objects, and the amount of detail.

Map series

It is convenient to identify a series of paper maps by their scale (the 1:50,000 water atlas), or the amount of earth they cover (e.g. NTS 2-degree letter blocks). Neither of these is well suited to GIS data. GIS data can be displayed at any scale, and can be manipulated as a seamless coverage for either analysis or display. GIS coverage should be identified by its accuracy (or uncertainty) and data resolution or density (or minimum feature size).

###### MAP SCALE - Notes and Chart
 Scale Miles/inch Line Width on Ground* Examples 1:2,000,000 ~32 2000’ USGS Nation-wide maps 1:1,000,000 ~16 1000’ National and state maps 1:500,000 ~8 500’ State or regional maps 1:250,000 ~4 250’ US Army Map Series 1:100,000 ~1.6 100’ USGS 30' quads 1:62,500 5208 feet 62.5’ USGS 15' quads 1:24,000 2000 feet 24’ USGS 7' quads 1:15,840 1320 feet 15.84’ Soils 1:9,600 800 feet Aerial photos

*Approximate real width on ground of a pencil line on a map - harder pencils give a finer line

1:2,000,000 to about 1:250,000 are SMALL-scale maps. Items on these maps appear small (e.g. a county on a 1:2,000,000 map is much smaller than on a 1:100,000 map).

1:24,000 on towards 1:9,600 are LARGE-scale maps. Items on these maps appear larger. 1:100,000 are pretty much in the middle - intermediate scale.

Other Reference Sites:

http://mapping.usgs.gov/

http://idaho.usgs.gov/reference/map_scales.html

http://mac.usgs.gov/mac/isb/pubs/factsheets/fs03800.html

http://www.usgs.gov/education/learnweb/MA/MAlesson7.html

http://www.epa.gov/ceisweb1/ceishome/atlas/learngeog/understandingmaps.html