msuheader2.gif (15254 bytes)Topographic Map
Interpretation

TECHNIQUE NUMBER 9 area.gif (2781 bytes)

AREA

X and Y axes

The measurement of area is one of the seldom-used strengths of quantitative map analysis. Because a topographic map is a true vertical projection ("orthographic"), areas are accurately depicted. You need only a technique, of which there are several. The techniques run the gamut from crude to sophisticated, from cheap to expensive, from cut-and-weigh to computer. All of the techniques have one aspect in common - comparison of measured data to a standard area. The standard area is a known area at the map scale. For example, at a map scale of 1:63,360, one square inch on the map equals one square mile on the ground. At a map scale of 1:24,000, one square centimeter on the map equals 57,600 square meters on the ground. For some of the techniques a standard area will have to be scaled off and measured. It is CRITICAL that this be done with the utmost accuracy. If it is not, a consistent error, or bias, will be introduced.

 

Cut-and-Weigh. One of the simplest techniques is the cut-and-weigh method.

  1. The area to be measured (and a standard area) are photocopied.  It is important that both the measured and standard areas be photocopied, as most copiers change scale slightly and would affect accuracy (introduce a bias) if only one shape is copied.
  2. Both the unknown and standard areas are cut out.  It is not critical to cut exactly on the edge of the area to be measured, as random variation on either side of the line will tend to cancel. You should try to avoid a bias to one side or the other.
  3. The shapes are weighed on an analytical balance with microgram (0.001 g) accuracy.
  4. The weight of the unknown area is divided by the weight of the standard, then multiplied by the area of the standard to determine the unknown area.

This method may sound crude, but it has been used in practice with a precision (reproduceability) of better than 5%.

cutweigh.jpg (10581 bytes)
Graph Paper. Another common method of determination of area is to compare the area of the area to the area of a measured number of precisely drawn squares (graph paper).
  1. Lay a sheet of nearly transparent graph paper over the area to be measured.
  2. First count the graph paper squares which fall entirely within the area to be measured.
  3. Then count the squares which are partially (NO MATTER HOW MUCH OR HOW LITTLE) within the area to be measured. Multiply this number by 0.5 (because the average partial square is half-filled).
  4. Add the result from step 3) to that from step 2) to get the approximate number of ful squares.
  5. Multiply by the known area of a full square.

The accuracy of this method depends on three variables: the consistency with which partial squares are counted, the size of the graph paper squares, and any bias which might be introduced by using a square grid to measure a square or rectangular area. To get around this last bias there are commercially available counting grids which use randomly distributed dots rather than regular squares. For the measurement of irregular areas, however, a square grid works fine.

graphcount1.gif (6618 bytes)
graphcount2.gif (3243 bytes)
Compensating Polar Planimeter. There is a relatively inexpensive gizmo called a compensating polar planimeter, which is readily available and can measure irregular areas rapidly and accurately. It consists of an anchor with a hinged rotating arm on which a counter and tracer point attached to a wheel are mounted. Although the counter both adds and subtracts as the tracer point is moved clockwise around the irregular shape, the net change is positive and is indicative of the area traced. These instruments have standard scales for standard maps, or a standard area can be traced.
  1. To use such an instrument, trace continuously around the standard area 5 or 10 times (the machine will continue to accumulate "area").  Be sure that you close on the point at which you started.
  2. Divide the figure on the counter by the number of times you traced it in order to come up with a count for the standard area. This process minimizes random error (fluctuations to either side of the line), but does not address bias.
  3. Divide the count by the area of the standard to determine the scale factor (counts per unit area - km2, miles2, hectares, ...)
  4. Trace clockwise (counterclockwise subtracts area) around your unknown area (several times if desired). Again, be sure that you close on the point at which you started.
  5. Divide the unknown area by the scale factor to determine the actual area.

NOTE:  Dr. Locke has several of these, which may be checked out.

Digitizer. There are tablets and software available for microcomputers which allow areas as well as lengths and locations to be instantly generated from a map. If you have one available to you, try it out!

planimeter.GIF (2863 bytes)

digitizer.gif (3123 bytes)

 

EXERCISE: Using any of the methods outlined above, calculate the area of elevated ground between the floodplains of Hyalite Creek and Cottonwood Creek and the Gallatin Range front on the Bozeman 15' quadrangle.

__________________ km2
Copyright 1998 by W. Locke.  Commercial use prohibited.  Educational use permitted under condition of a courtesy notification to Dr. Locke.  Return to Dr. Locke's home page.