TOPO Map scales

In extracting digital images from the TOPO map series, I came to grips with some interesting questions about map scales.

The 7.5 minute map series is claimed to be at a scale of 1:24,000 (and I do believe it). I have a couple of maps handy (paper originals from the USGS) and took it upon myself to measure the size of the actual map image.

The O'Neill Hills Quadrangle (in SW Arizona) is at a Longitude of 113:15 and a latitude of 32:00 (at the SE corner of the quad and coordinates given as degrees:minutes). The map image measures 19.312 inches wide and 22.625 inches high.

The North Palisade Quadrangle (in E central California) is at a Longitude of 118:30 and latitude of 37:00. The map image measures 18.156 inches wide and 22.625 inches high.

It is no surprise that a map at a more northern latitude would be narrower, as the lines of constant longitude get closer together as one moves north, eventually converging at the pole. Let's see if these measured sizes make sense given what we know about the size of the earth.

An accepted value for the radius of the earth at the equator is 6378.14 km (3963.19 miles = 251107870 inches). 1 arc-minute is 0.000290888 radians and would subtend 73044.3 inches on the surface of the earth. At a scale of 1:24000 on the map, this would be 3.0435 inches, and 7.5 minutes would be 22.82 inches.

This is a pretty good match for the height of a USGS 7.5 minute map, but we are off by 0.2 inches, lets back calculate what value for the radius of the earth they must be using. 22.625 inches of height would represent 543000 inches on the ground, so 1 arc minute would be 72400 inches on the ground and would give a radius of 248882777.6 inches, which is 6321.88 km. Even given the fact that the earth's globe is not a perfect sphere and is flattened at the poles this is a bit odd, the radius at the pole is 6356.7 km given the accepted flattening of 1/298.257. My guess is that my paper maps have shrunk from 22.82 inches to 22.625 inches (an amount of 0.86 percent), which is entirely within the realm of reason.

Now, let's see if we can calculate the expected width of a 7.5 minute maps given the latitude. This is no great trick, simply the cosine of the latitude. At 32 degrees this is 0.848 and so the width of a 7.5 minute quad (given my shrinkage) is 19.19 inches (close to the 19.32 measured for O'Neill Hills). At 37 degrees this is 0.799 and so the shrunken width would be 18.069 inches (close to the 18.156 measured for North Palisade). And we don't expect paper to shrink equally in both X and Y directions, so we can account for the discrepancies here as well.