DMS to Decimal Degrees Converter
DMS to decimal is degrees plus minutes over 60 plus seconds over 3600, worked in exact fractions, with the hemisphere letter turned into a sign and the decimals matched to the input.
Decimal degrees
- How it was read The letter S makes this a latitude.
- 33° 51' 54.5", south (degrees, minutes and seconds)
- Decimal degrees Signed: negative is south of the equator or west of Greenwich. This is the form mapping software and spreadsheets expect, with no letter. 6 decimal places of a degree is 11.1 cm at the equator.
- -33.865139
- Degrees, minutes, seconds Rounded once, in units of the last second shown, and split afterward, so a value just under a whole minute carries into the minute instead of printing as 60". The letter carries the sign; a minus is not written as well.
- 33° 51' 54.50" S
- Degrees and decimal minutes The form most GPS receivers, chartplotters and marine charts use. A minute of latitude is about one nautical mile, which is why navigators keep the minutes.
- 33° 51.9083' S
- The arithmetic Degrees + minutes/60 + seconds/3600, done in exact fractions. An ellipsis means the decimal never ends, which is the usual case for anything with seconds.
- 33 + 51/60 + 54.5/3600 = 33.865138888888...
- What the input can resolve To keep that and invent nothing, write 5 decimal places of a degree, 1 of a second, or 3 of a minute.
- the nearest 0.1 seconds, about 3.09 m
- A degree here On the WGS 84 ellipsoid at this latitude. Lines of longitude converge toward the poles, so a degree of longitude shrinks with the cosine of the latitude while a degree of latitude stays near 111 km. At this latitude the 6th decimal of a longitude is 9.3 cm.
- 110.92 km north-south, 92.53 km east-west
| Decimal places | One step | At the equator | Same step in seconds | East-west at this latitude |
|---|---|---|---|---|
| 0 | 1° | 111.32 km | 3,600" | 92.53 km |
| 1 | 0.1° | 11.13 km | 360" | 9.25 km |
| 2 | 0.01° | 1.11 km | 36" | 925.31 m |
| 3 | 0.001° | 111.32 m | 3.6" | 92.53 m |
| 4 | 0.0001° | 11.13 m | 0.36" | 9.25 m |
| 5 | 0.00001° | 1.11 m | 0.036" | 92.5 cm |
| 6 | 0.000001° | 11.1 cm | 0.0036" | 9.3 cm |
| 7 | 0.0000001° | 1.1 cm | 0.00036" | 9.25 mm |
| 8 | 0.00000001° | 1.11 mm | 0.000036" | 0.93 mm |
- Distances use the WGS 84 ellipsoid (EPSG::7030): equatorial radius 6,378,137 m exactly, inverse flattening 298.257223563. One degree along the equator is 2 x pi x 6,378,137 / 360 = 111,319.49 m.
- GPS.gov states that GPS-enabled smartphones are typically accurate to within a 4.9 m radius under open sky, so past the fifth decimal of a degree a phone reading is reporting digits, not position.
- Marks are written with the ASCII apostrophe and double quote rather than the prime symbols, because that is what pastes cleanly into mapping software and spreadsheets.
How the Degrees Minutes Seconds (DMS) Converter works
The formula is decimal degrees = D + M/60 + S/3600, and then a minus sign if the hemisphere is south or west.
Worked through for 33° 51' 54.5" S: 51 minutes is 51/60 = 0.85 of a degree. 54.5 seconds is 54.5/3600 = 0.0151388... of a degree. Add them to 33 and you get 33.8651388..., and the S makes it -33.8651388... . That is the number a mapping tool, a spreadsheet or a GPS track file expects, and the reason the letter becomes a sign is that decimal coordinates have nowhere else to put it: north and east are positive, south and west negative, with zero at the equator and at the Greenwich meridian.
Two things go wrong in this direction. The first is the sign. Drop it, and the point moves to the other hemisphere. A coordinate in Sydney turns up in the North Pacific, which is easy to spot, but one near the equator or the prime meridian lands a short distance away and looks plausible. If the source has a letter, keep it attached to the value you paste, and check any column of converted coordinates for the expected sign before you trust a map.
The second is how many decimals to keep. The input above is given to a tenth of a second, which is about 3.1 m on the ground at the equator. A tenth of a second is 1/36,000 of a degree, so it takes five decimal places of a degree (about 1.1 m) to hold it without losing anything. A sixth or seventh place adds digits that the original measurement never had. Keeping only three or four places goes wrong the other way and throws away real information: four places is about 11 m.
The page does the division in exact fractions on the digits you typed, which matters more than it sounds. 54.5/3600 has no exact binary representation, so ordinary floating-point arithmetic gives a value off in the seventeenth significant figure. That is far below anything physical, but it is the source of the stray ...0000001 and ...9999999 tails that appear when a spreadsheet converts thousands of coordinates and someone later compares them as text.
The input can be written almost any way: 33 51 54.5 S, S33° 51' 54.5", 33:51:54.5 with a minus in front, or with the curly primes that word processors substitute. It also takes degrees and decimal minutes from a GPS (33° 51.908' S) and converts those the same way, minutes over 60. If the reading is not what you meant, the first row of the result says exactly how each number was interpreted.
Degrees Minutes Seconds (DMS) Converter FAQ
What is 33° 51' 54.5" S in decimal degrees?
-33.865139 to six places. 51/60 = 0.85, 54.5/3600 = 0.015139, and 33 + 0.85 + 0.015139 = 33.865139, negative because S is south of the equator.
Do south and west always become negative?
In decimal degrees, yes. That is the convention used by GPS exchange formats, web maps and the ISO 6709 standard for coordinate strings: positive is north and east, negative is south and west.
Can I convert a whole column of DMS values?
This converts one value or one latitude-longitude pair at a time. In a spreadsheet the same formula is =D+M/60+S/3600, multiplied by -1 when the letter is S or W; the decimals advice above applies there too.
How many decimal places does one second of arc need?
Four. One second is 1/3600 of a degree, about 0.000278, and four decimal places (0.0001°, about 11 m) is the fewest that hold it. A second itself is about 31 m at the equator.