Planimetric versus Surface Area Calculations

Paul Sutton, Associate Professor, Geography, University of Denver

Mario Lopez, Associate Professor, Computer Science, University of Denver

Last month’s Beyond Mapping column (GeoWorld, December 2002, see link below) on the difference between planimetric and surface area got us thinking—what would be the shape of Colorado if you ironed out the winkles of the Rockies?  Would there be a dramatic difference between the actual surface area and the planimetric area represented by traditional maps?  How would you generate and display the maps necessary for such an analysis?

Figure 1.  Elevation Map of Colorado.

Figure 1 is a display of the readily available 90 meter DEM data for the state.  The green tones identify lower elevations of the eastern plains and western slope that transition to white areas of the Rocky Mountains in the central portion.  In a sense, you can think of the state as a dirty shirt with a bunch of winkles in the middle.  The first step in ironing-out the wrinkles is to calculate a slope map of the area.  We used ArcGIS’s grid command Slopegrid = slope(CO90mDEM, degree).

The next step calculates the surface area for each 90 meter grid based on the location’s slope using the equation…

Surface Area = (90m)2 / cos(slope angle)

…and expressed as SurfArea = int(PlanArea / cos(SlopeGrid div deg) in ArcGIS.

Figure 2.  Slope Map of Colorado.

Figure 2 shows the results for the surface area calculations with white indicating 0 degrees (no change in area) through dark red for 86 degrees (tremendous change).  The calculation for this extreme condition is grid cell area / cosine (slope angle)= .81ha2 / cosine (86)= 11.6ha2.  Note that the biggest changes are in the mountainous areas as you would expect.  Also note what appears to be a relationship between county boundaries and large changes (steep areas)—very interesting.

The final step summarizes the results for each county using the region-wide summary command (Summarize Zones in ArcView).  This procedure calculates the average percent change for each of the sixty-five counties in Colorado.

Figure 3.  Percent Change in Area by County.

Figure 3 shows the results with the light-toned eastern counties having zero percent change, medium-toned from 1-4 percent change  and the dark red-toned from 4.6 to 9.3 percent change.  The seven darker cross-hatched counties exhibit the greatest change from 4.6% through 9.3%. San Juan County in the southwestern portion of the state had the highest change 9.3%.

The inset in top-left portion of the figure compares the planimetric representation of San Juan County (red) and its surface area representation (green).  The backdrop of elevation confirms that the county certainly is mountainous.

The ironing out of Colorado proved to be a lot less dramatic than we expected.  While some of the more rugged terrain exhibited considerable difference in area, they were far out numbered by areas of gentler slopes having minimal differences.  Also, while the Rocky Mountains might seem steep, they rarely have slopes exceeding sixty degrees.  The cosine relationship used in surface area calculation is exponential with big difference kicking in at very large slope angles.

The exercise, however, is a good one for conceptualizing the differences between planimetric and surface area representation.  Come to think of it, would the Canadian Rockies around Banff exhibit a greater difference in area?  Or would the Himalayas or Andes hold record differences when ironing out the terrain?  Sounds like an interesting project.

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Note:  The referenced Beyond Mapping column is posted online at

http://www.innovativegis.com/basis/MapAnalysis/Topic11/Topic11.htm, select “Calculating Realistic Areas”

For background theory and equations on calculating surface area and surface length and inclination of a line, see…