Relating Latitude and Longitude to Everyday Distances

I’m a cartography fan and Wikipedia’s geographic coordinates add another level of utility to articles.   Many of the latitude /longitude notations use six decimals of precision though which seemed like overkill.  When I recently added a {{coord}} template to an article,  I calculated how much distance each digit represents using imperial units (those familiar with metric units can look here).  Briefly, each tick of the sixth place changes position by about 8.5 cm or 4-⅜ inches (for latitude; longitude changes are even smaller and vary with the cosine of latitude).   Consumer GPS will only be accurate to about four decimals (or maybe five with WAAS).

The detailed calculation starts with the nautical mile to relate lat/lon coordinates to everyday distances.  HowStuffWorks.com explains

“The length of a nautical mile is based on the circumference of the planet Earth. If you were to cut the Earth in half at the equator, you could pick up one of the halves and look at the equator as a circle. Then divide that circle, or arc, into 360 degrees and divide each degree into 60 minutes. A minute of arc on the planet Earth is 1 nautical mile. This unit of measurement is used by all nations for air and sea travel.”

Most of us don’t normally use nautical miles to pilot ships or airplanes and instead rely on smaller units like kilometers and miles.   For the conversion examples below, I’ll use the coordinates of the entrance to the USGS’ office in Menlo Park :

37° 27′ 22.86″ N,   122° 10′ 15.6714″ W
37.456350         , -122.17102

The notation degrees° minutes seconds  converts to decimal notation as shown here and then we can calculate typical distances corresponding with each position in the notation:

37 + 27/60 + 22.86/3600 = 37.456350
37°  27'     22.86"
 |    |       |  |
 |    |       |  +- 1/100 of an arcsecond = 1.0127 feet
 |    |       +---- One arcsecond    =  1 nautical mile / 60 = 101.27 feet
 |    +------------ One arcminute    =  1 nautical mile  = 1.15077945 statute miles
 +----------------- One (arc) degree = 60 nautical miles = 69 statute miles

In the decimal notation for this latitude, each position corresponds to the following north/south travel distances:

37.456350
 | ||||||
 | |||||+- 0.00006 naut mi. = 0.3646 feet= 4.3748 or 4-3/8 inches=11.112 cm
 | ||||+-- 0.0006 naut. mi. = 3.646 feet =   1.1112 m
 | |||+--- 0.006 nautical mi= 36.46 feet =  11.112 m
 | ||+---- 0.06 nautical mi.= 364.6 feet = 111.12 m
 | |+----- 0.6 nautical mi. =  0.69 statute miles=   1.1112 km
 | +------ 6 nautical miles =  6.9 statute miles =  11.112 km
 +------- 60 nautical miles = 69 statute miles   = 111.12 km

Longitude lines. Traveling N/S along one changes your latitude.

(For reference, 1 nautical mile is defined as exactly 1.852 km and is about equal to 1.1508 statute miles.)

Traveling one arcdegree of latitude  north/south (by moving along a longitude line) represents the same distance anywhere on earth.

One degree of E/W travel requires different distance at different latitudes.

Traveling one arcdegree of longitude east/west (by moving along a latitude line),  however,  represents different distances depending on the cosine of the latitude’s degrees (north or south).  At higher latitudes, traveling one arcdegree east/west covers a smaller distance because circles of latitude are smaller at higher latitudes.

For example, in Caracas at 10 degrees north latitude, traveling one arcdegree along that latitude line represents east/west travel of:

1 arcdegree = 1 naut. mi. * cos(10 deg) = 0.985 nautical miles = 1.333 statute miles

but at 37.45635 degrees north, 1 arcdegree of east/west travel represents just:

1 arcdegree = 1 naut mi. * cos(37.45635 deg) = 0.7938 nautical miles = 0.9135 statute miles

These calculations are close to correct, however, the Earth is not quite a perfect sphere and so professional geodetic measurements use a mathematical model of those imperfections called a datum.  For more conversions, and pointers to the nuances introduced by geodetic systems, see http://transition.fcc.gov/mb/audio/bickel/DDDMMSS-decimal.html .

Without access to professional surveying equipment and geodetic correction software, five decimals or 1/10 of an arc second, representing around 1m of accuracy, is about as accurate as a consumer GPS is likely to measure.

Go hiking and enjoy!

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