HOW DO YOU CHANGE A DECIMAL TO A DEGREE TO GET A PERCENTAGE

Table 1. Latitude / longitude coordinates (expressed in degrees, minutes and seconds) of four cities
City	Latitude	Longitude
Acapulco	16° 58' 26.4" North	99° 55' 55.2" West
Oaxaca	16° 55' 15.6" North	96° 56' 31.2" West
Buffalo	42° 53' 56.4" North	78° 50' 52.8" West
Detroit	42° 23' 38.4 North	83° 4' 44.4" West

Table 2. Latitude / longitude coordinates from Table 1 expressed in decimal degrees.
City	Latitude	Longitude
Acapulco	16.974	-99.932
Oaxaca	16.921	-96.942
Buffalo	42.899	-78.848
Detroit	42.394	-83.079

In contrast, Table 3 shows the coordinates of these same four cities using the truly Cartesian Universal Transverse Mercator (UTM) spatial coordinate system (the UTM system will be described in detail later in this learning guide; for now, all you need to know about the UTM system is that it is Cartesian).

Table 3. Universal Transverse Mercator (UTM) coordinates of four cities
City
Acapulco	1,876,843	400,830	14
Oaxaca	1,871,838	719,190	14
Buffalo	4,751,593	675,667	17
Detroit	4,695,438	328,887	17

By applying the Pythagorean theorem to first the decimal degree version of the latitude / longitude coordinates shown in Table 2 and then to the UTM coordinates from Table 3, we can compute two different estimates of the distances between the two U.S. and the two Mexican cities. These distances are shown in Table 4. Also shown in Table 4 are the surveyed distances between these pairs of cities, which are (presumably) quite accurate. Finally, Table 4 shows the difference in distances between the U.S. and Mexican cities, both as percentages and as deviations from surveyed distances.

Table 4. Distances Between U.S. and Mexican Cities
City Pair	Distance Computed Using Latitude / Longitude Coordinates	Distance Computed Using UTM Coordinates	Actual Distance
Acapulco - Oaxaca	2.99 Degrees	318,399 Meters	318,029 Meters
Buffalo - Detroit	4.26 Degrees	351,297 Meters	350,484 Meters
Percent Difference	42.49%	10.33%	10.21%
Deviation from Surveyed Percent Difference	32.28%	0.12%	0.00%

Table 4 shows that when surveyed, the distance between the U.S. cities is 10.21% greater than the distance between the Mexican cities. The Cartesian UTM coordinates produced very nearly this same difference (10.33%, a deviation of only 0.12% from the surveyed difference). However, the distances computed using the non- coordinates differ by over 42%, which is a deviation of over 32% from the surveyed difference in distances.

What causes the latitude / longitude coordinates to produce distances estimates that are so wildly out of line with surveyed distances, while the UTM system produces estimates that seem so much more accurate? Recall that one of the characteristics of a Cartesian coordinate system is that the units of measure are the same along the entire length of the axis. The UTM system conforms to this characteristics; distances along both axis are measured in meters. But in the latitude / longitude system, the basic unit of measure is a degree, and the distance that a degree represents is not constant. Consider a degree of longitude. As a very rough average, its about 24,900 miles around the equator. There are 360 degrees of longitude around the equator, so each degree of longitude must cover 24,900 ÷ 360 = 69.167 miles. Now consider a degree of longitude at the North Pole. At the pole, you can pass through all 360 degrees of longitude by simply turning around. This means that at the North Pole, each degree of longitude must equal 0 ÷ 360 = 0 miles.

This explains why the latitude / longitude distance estimates shown in Table 4 are so odd -- the units they are based upon -- degrees -- are not constant. The degrees of distance separating the Mexican cities are longer than the degrees of distance separating the U.S. cities. Figure 2 makes this clear. Figure 2 shows the map from Figure 1, but a graticule has been added. This graticule shows lines of latitude and longitude 10 degrees apart. Note how the lines of longitude converge toward the poles, thereby shrinking the distance between the lines. Since the lines of longitude are always separated by 10 degrees, the only way they can converge is if the length of each degree decreases.

Figure 2. Map from Figure 1 with a graticule added (Lambert projection).