Scale


Maps have to show big areas of the earth on small pieces of paper. To make everything fit, objects on the surface of the earth, like a house or a road or a mountain, must be drawn much smaller than they really are. In order to use the map to determine, for example, how far it is from one town to another, we have to know how much smaller the map is than reality. The amount of reduction is called the map "scale." It is very important to know the scale of the map you are using. For example, say that you want to go for a hike between two places that are only one inch apart on a map. On a map like your Cumberland quadrangle, the distance would be less than 1/2 a mile. However, if you were looking at a world map, those two places might be hundreds or thousands of miles apart.

Scale can be stated in three ways. Two of these different kinds of scale are shown on your map. The three ways of stating scale are the verbal scale, the graphic scale, and the representative fraction. The graphic scale is sometimes called the bar scale, and the representative fraction is the RF or the fractional scale.

Verbal Scale

The verbal scale is not shown. It is a simple statement of the scale designed to directly communicate the size relationship between the map and the real earth. An example is one inch represents one mile or 1" = 1 mile. This means that one inch on the paper represents one mile on the earth. A good verbal scale would have one unit on the map. Anything other than this would need to be converted to be useful. The unit applied to the map should also be a useful or appropriate unit for that map. Since most maps fit in books or on the tables in front of us, these units would be inches, centimeters, or millimeters. Finally, the distance on the earth should be in the simplest form. The goal is communication, so if this unit does not readily communicate, the scale is improperly converted. The ground distance must be meaningful.

Graphic Scale

Another form of the scale is the graphic scale or bar scale. Three different bar scales can be found near the bottom center of your map. They look like straight lines, of varying lengths, marked off with numbers. One of these bar scales is divided into kilometers (1000 meters in a kilometer, 100 centimeters in a meter), one into miles (5280 feet in a mile) and one into feet (12 inches in a foot). Any of these bar scales can be used to relate sizes or distances as measured on the map to sizes and distances as they are found on the earth's surface. For example, say you want to know how many feet it is, in a straight line, from the start of the bridge over the C & O Canal through Knobly Tunnel to the road on the west of Knobly Mountain just to the west of South Cumberland. Take a piece of paper with a smooth, straight, fresh edge. Lay that edge on the map so that it runs across the stream, along the rail line through the tunnel and across the road. Make a pencil mark on the paper to show the location of the edge of the canal at the start of the bridge, and one at the road on the west side of the mountain. Now look at the bar scales and find the one labeled in feet. Note that the zero is not located on the end of the bar on any of these scales. Now, lay the edge of your marked paper along the bar that is scaled in feet. Move the right hand mark to one of the subdivisions so that the left hand mark falls between the zero and the left end of the scale. To the right, the bar is marked off in 1000s of feet. To the left, the divisions are 200 feet. It is likely that the left hand mark will not fall exactly on a number. In that case you will have to estimate the distance. The number where the right hand mark falls, a specific distance, is added to this small estimated distance to give you the total distance between these two points in feet. For example, if the right hand mark is positioned on the 4000 mark and the left hand mark falls 1/2 of the way between the left end of the scale and the last little subdivision, the distance would be 4000 plus 900 or 4900 feet.

How far is it from the start of the bridge over the C & O Canal through Knobly Tunnel to the road on the west of Knobly Mountain just to the west of South Cumberland? Answer

Now, using the same marks on the piece of paper, but a different bar scale, determine the distance between the two points in kilometers. Again, pay attention to the location of the zero on the bar. Answer



Remember that if you are measuring a distance longer than the length of the bar, that you must be sure to consider the location of the zero. For example, the length of the entire kilometer bar is TWO kilometers, and the feet bar is 8000 feet long.

The graphic scale is the best scale to use if the map is going to be reproduced and changed in size because both change at the same ratio, and the relationship between the map and the earth will be maintained. Enlargement or shinkage make the other two scales erroneous! This is an important consideration and problem when using maps on computers. Only a graphic scale as part of the map will remain functional as you change the view by zooming in or out.

To measure a curved feature like a stream or an interstate, you again use the edge of a sheet of paper. Lay the edge along the line from the starting point, and where the road or stream deviates from the edge of the paper, make a light pencil mark on the paper and on your map. Then turn the edge of the paper so it lines up again along the road keeping the mark on the edge of the paper and the mark on the map exactly lined up. Continue doing this for the length of the distance to be measured. This is like stretching the curved line out along the edge of your paper. Once you have the two end points determined, you can then use the graphic scale to determine what distance that length represents.

Remember, as long as you have a bar graph, you should use the edge of a sheet of paper to make your measurements. You should not use a ruler or a piece of string. A ruler will not allow you the accuracy of a sheet of paper and a sharp pencil, and a string is less stable and tends to stretch.

Representative Fraction

Scales compare, and convert, the distance measured on a map to the real distance found on the earth. Representative fractions are a mathematical method for making this conversion. Representative fractions are by far the most useful and most important method for calculating distances on maps, so make sure you understand their use! The representative fraction of a map is usually shown as the number 1 separated by a colon from another number. Look near the bottom center of your map to find the representative fraction. It is 1:24,000. This RF can also be written as a fraction like 1/24000 or with the horizontal line rather than the forward slash. The first number represents a distance measured on the map. The second number converts the first number to the actual distance on the earth.

For example, say that you want to make the same measurement as above, the distance through Knobly tunnel, and no graphic scale is given. First, you have to measure the map distance. Take out a ruler to measure how far it is between these two points on the map. The map distance is about one inch. To convert map distance to the actual distance you use the representative fraction 1:24,000. This fraction means that 1 inch (or foot, or meter, or mile, or whatever unit you use) is equal to 24,000 inches (or whatever unit) on the surface of the earth. In this case, since these points are one inch apart on the map, it means they are 1 time 24,000 inches apart. If it had been two inches it would have equaled 48,000 inches apart on the surface of the earth. These numbers do not readily communicate, so we want to simplify them and convert to feet. To do so, divide 12. 24000 inches is 2,000 feet. 48,000 inches would be 4,000 feet.

Say you want to find out how far it is from one point to another and the distance measures 3 1/2 inches. To determine the actual distance you would take 3.5 times 24,000 and so discover that the distance is 84,000 inches. Simplifying this you find that it is 7000 feet or about 1 1/3 miles. (Note, the first number is mathematically accurate, the second was an approximation, but all measurements are inexact or involve some level of approximation.)

To summarize the RF, three rules must be maintained. First, the numerator is always one. Second, both the numerator and the denominator represent the SAME UNIT of measurement. What this unit is does not matter, but both are the same. This provides the mathematical relationship of unit/unit which is equal to 1, and thus, drops or cancels out of the relationship. Third, a representative fraction NEVER has a unit of measurement given with it. The beauty of the RF is that what ever measurement system you wish to use can be applied. If your pet turtle ended up on your map fitting exactly between the two points you are measuring between and the scale was 1:24000, that means that you could place your turtle between those two points on the ground 24000 in exactly that same orientation. With the representative fraction, it is very helpful to know the number of inches in a mile. This MAGIC NUMBER is 63,360. 1 mile equals 63,360 inches. A common RF scale is 1:63360 or 1:316,800. What do these mean?

Answer

Scale Comparisons

Consider the size of the fractions 1/100 and 1/2. The fraction 1/100 is a much much smaller number than is 1/2. As the denominator gets larger, the fraction gets smaller. The scales of maps can be compared using the RF. A large scale map shows limited distance, but is able to show more detail. A small scale map shows larger distance, but must have more generalization and contains less details. A topographic map is generally considered to be a large scale map, although if it were compared to a map of campus or an architect's building plan, it would probably be at a smaller scale. A world map, with an RF such as 1:1,000,000 is a small scale map.



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