How to convert decimal degrees to degrees, minutes and seconds (and back)

Coordinates come in two common formats. A GPS app, a spreadsheet or the Everyday Tools address lookup gives decimal degrees, such as 38.84506, -76.92837. A deed, a survey plat or an older chart gives degrees, minutes and seconds, such as 38° 50' 42" N, 76° 55' 42" W. Both describe the same point, and converting between them is simple arithmetic once you know that a degree splits into 60 minutes and a minute into 60 seconds [1].

What the three parts mean

Latitude runs from 0 at the equator to 90 degrees at either pole [2]. Longitude runs from 0 at the prime meridian through Greenwich, England, to 180 degrees on the far side of the Earth [3]. Each degree is divided into 60 minutes and each minute into 60 seconds, just like an hour of time [1][2]. The standard symbols are ° for degrees, ′ for minutes and ″ for seconds [1].

The letters N, S, E and W name the hemisphere. Decimal degrees drop the letters and use a sign instead: north and east are positive, south and west are negative. The mainland United States is west of Greenwich, so its longitudes are negative in decimal form.

Degrees, minutes and seconds to decimal degrees

Divide the minutes by 60, divide the seconds by 3,600, and add both to the degrees. Take 25° 46' 27" N. The minutes give 46 divided by 60, which is 0.76667. The seconds give 27 divided by 3,600, which is 0.0075. Add them to 25 and you get 25.77417, positive because the direction is north. For 80° 11' 37" W, the minutes give 0.18333, the seconds give 0.01028, and the total is 80.19361. Because the direction is west, the final value is -80.19361.

Write down the degrees as the whole number part.
Divide the minutes by 60 and add the result.
Divide the seconds by 3,600 and add that as well.
If the direction letter is S or W, make the number negative. Then drop the letter.

Decimal degrees to degrees, minutes and seconds

Going the other way, peel off the whole degrees, multiply what is left by 60 to get minutes, then multiply what is left again by 60 to get seconds. When we looked up the sample address in the Everyday Tools address to coordinates tool, 4600 Silver Hill Rd, Washington, DC 20233, the Census geocoder returned latitude 38.84505589808 and longitude -76.92836638093.

For the latitude, 38 is the degrees. The fraction 0.84505589808 times 60 is 50.70335, so there are 50 minutes. The remaining 0.70335 times 60 is 42.20 seconds: 38° 50' 42.20" N. For the longitude, set the minus sign aside and work with 76.92836638093. The degrees are 76, the fraction times 60 is 55.70198, so 55 minutes, and 0.70198 times 60 is 42.12 seconds. The minus sign becomes a W: 76° 55' 42.12" W. Minutes and seconds always land between 0 and 59; if you get 60 or more, you forgot to subtract the whole number before the next step.

A third format: degrees and decimal minutes

Many handheld GPS units and marine charts stop halfway, giving degrees and decimal minutes: 38° 50.703' N. Divide the minutes by 60 and add them to the degrees: 38 plus 50.703 divided by 60 is 38.84505. Sailors like this format because one minute of latitude is one nautical mile, 1.1508 statute miles or 1.852 kilometers [4].

How many decimal places to keep

One second is 1 divided by 3,600, or about 0.00028 degrees. Four decimal places cannot quite hold whole seconds: rounding the latitude above to 38.8451 converts back to 42.36 seconds, while 38.845 gives exactly 42. Five decimal places, as in 38.84506, keep seconds to within a fraction of a second.

On the ground, a second of latitude covers about 30.7 meters [2]. At 38 degrees north, the U.S. Geological Survey puts a second of latitude at 101 feet and a second of longitude at 80 feet [5]. Longitude shrinks toward the poles: a degree covers about 111 kilometers at the equator, 56 kilometers at 60 degrees north or south, and nothing at the poles [3].

Mistakes that move the point

Typing minutes and seconds as if they were decimals. Entering 38° 50' 42" as 38.5042 instead of 38.845 is an error of 0.34 degrees, which at about 69 miles per degree of latitude [5] is roughly 23 miles.
Dropping the minus sign on a west longitude. 76.93 without the sign is a point in the Eastern Hemisphere, on the far side of the world.
Swapping the order. Most writing puts latitude first, but mapping software often stores longitude first as x and latitude as y; the Census geocoder returns x for longitude and y for latitude [6]. A U.S. point with a latitude near 77 and a longitude near 38 has been swapped.

Using the result with the address tool

The Everyday Tools address to coordinates tool works in decimal degrees only. The forward lookup, Address → coordinates, shows Latitude and Longitude as decimals, and Copy coordinates copies them as latitude, longitude. The reverse lookup, Coordinates → address, asks for decimal degrees in the Latitude and Longitude fields, with a minus sign for south or west. Paste 38°50'42" into the latitude field and the tool stops you with Enter latitude between -90 and 90 in decimal degrees. Convert first, then press Find nearby address.

Sources

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