How do you convert DMS to decimal degrees?

Formula, assumptions, and limits

The position 33° 51′ 32.4″ S, 151° 12′ 23.4″ E is −33.859000, 151.206500 in decimal degrees and 33° 51.540′ S, 151° 12.390′ E in degrees and decimal minutes. One formula and its inverse carry every conversion on this page.

Formula

• $DD$: decimal degrees, from −90 to +90 for latitude and from −180 to +180 for longitude.
• $D$: whole degrees, 0 to 90 for latitude and 0 to 180 for longitude.
• $M$: minutes of arc, at least 0 and under 60. In degrees and decimal minutes, $M$ carries the fraction and $S$ is 0.
• $S$: seconds of arc, at least 0 and under 60; the seconds of the output are $T - 3600D - 60M$.
• $s$: +1 for north or east, −1 for south or west.
• $T$: the absolute value in seconds, rounded to hundredths of a second, the last decimal shown, before it is split into parts.

Derivation

The BIPM SI Brochure, 9th edition (V4.01, June 2026), lists the degree, the minute and the second among the non-SI units of Table 8 and defines them in sequence: 1° = (π/180) rad, 1′ = (1/60)° and 1″ = (1/60)′. A second is therefore 1/3,600 of a degree, and the forward formula is the sum of the three parts in degrees. The sign comes from the hemisphere. ITU-R M.1371-6, the AIS standard approved in February 2026, writes latitude as ±90° with north positive and south negative, and longitude as ±180° with east positive and west negative, in Table 46.

The inverse undoes the sum. Rounding $\lvert DD \rvert$ to a whole count of the smallest unit shown, then taking whole degrees and whole minutes out of that count, keeps every part in range. Rounding the seconds on their own does not. For 10° 59′ 59.9999″, the seconds alone round to 60.00″ and print as 10° 59′ 60.00″, a form no chart or receiver uses. As a count it is 39,599.9999″, which rounds to 39,600.00″ and splits into 11° 00′ 00.00″.

Assumptions

• Both coordinates are on the same horizontal datum, and the result is on that datum too.
• A minus sign means south or west. A minus sign together with S or W is refused, so the sign cannot be applied twice.
• Zero latitude and zero longitude carry no hemisphere letter in the results: the equator is neither north nor south, and the Greenwich meridian neither east nor west. The A.851(20) line C group needs a letter, so it writes zero as 0000N or 00000E.
• With no hemisphere letters, the first coordinate is the latitude, so two bare numbers such as 33 51 are latitude 33° and longitude 51° E, not 33° 51′. A coordinate with no letter or sign is north or east.
• Outputs show six decimal places of a degree, three of a minute and two of a second.

Worked example

Take 33° 51′ 32.4″ S, 151° 12′ 23.4″ E.

1. Latitude: 33 + 51 ÷ 60 + 32.4 ÷ 3,600 = 33 + 0.85 + 0.009 = 33.859000; south makes it −33.859000.
2. Longitude: 151 + 12 ÷ 60 + 23.4 ÷ 3,600 = 151 + 0.2 + 0.0065 = 151.206500, positive for east.
3. Degrees and decimal minutes: 51 + 32.4 ÷ 60 = 51.540′ and 12 + 23.4 ÷ 60 = 12.390′, so 33° 51.540′ S, 151° 12.390′ E.
4. Back from decimal: 0.859 × 60 = 51.54, so 51′; 0.54 × 60 = 32.40″.

ISO 6709:2022 gives its own check in clause 3.19: 50.0795725 decimal degrees is 50° 04′ 46.461″. To two decimals of a second the calculator returns 50° 04′ 46.46″, and 50° 04.774′ in decimal minutes.

Edge cases and limits

Latitude stops at 90° and longitude at 180°, so 90° 00′ 01″ N is refused, as are 60 minutes and 60 seconds. Longitude 180° E and 180° W are the same meridian; the calculator keeps the letter that was chosen. A position in degrees and whole minutes, the form of the IMO resolution A.851(20) reporting line and of the ITU-R M.493-16 distress call, can be off by up to half a minute, about 0.5 nautical mile of latitude, against the position it was rounded from.

Regulatory basis

No IMO instrument prescribes how to convert a position between forms. The units are the BIPM’s, in Table 8 of the SI Brochure, 9th edition. The forms are set system by system: IMO resolution MSC.112(73), adopted 1 December 2000, section 3.1, has a shipborne GPS receiver show position on WGS 84 in degrees, minutes and thousandths of minutes; ITU-R M.1371-6 sends AIS positions as signed counts of ten-thousandths of a minute; and IHO S-4, Edition 4.10.0, B-131, writes chart positions in degrees, minutes and decimals of a minute with the minute tick after the fraction.

Common errors

• Reading the minutes as a decimal. 12° 30′ is 12.5°, not 12.30°; the 0.2° gap is 12 minutes of latitude, about 12 nautical miles. On this page’s default, writing 33° 51′ 32.4″ as 33.51324° moves the latitude 20.7 nautical miles.
• Dropping the sign of a south latitude or west longitude when pasting decimal degrees into software, which mirrors the position across the equator or the Greenwich meridian.
• Applying the sign twice, with a minus sign and a letter S, so that a tool that reads both flips the latitude back to north.
• Rounding the last part alone, which prints 60″ or 60.000′.
• Swapping the order. ITU-R M.493-16, Annex 1, section 5.3, Note 1, puts latitude first.

Which form each system uses

Each system on board takes its form from a different body: the IMO for GPS displays and ship reports, the ITU for AIS and DSC, the IHO for charts.

The AIS figures are the decimal degrees times 600,000. ITU-R M.1371-6 reserves 91° and 181° for “not available”, so a longitude field of 108,600,000 (181 × 600,000) means the ship sent no position. The DSC quadrant digits are 0 for NE, 1 for NW, 2 for SE and 3 for SW, and ten nines mean no position.

Decimal places and distance

A digit is worth a distance on the ground, and the form a position is written in says how fine it claims to be. Taking a minute of latitude as the 1,852 m nautical mile of the BIPM SI Brochure, Table 8:

The figures are nominal. On the WGS 84 ellipsoid, with semi-major axis 6,378,137 m and inverse flattening 298.257223563 as the US National Geospatial-Intelligence Agency publishes them, the length of a minute of latitude changes with latitude, so the nautical mile and the minute agree only approximately. Along a parallel the step shrinks with the cosine of the latitude: at 33.859° a degree of longitude is about 49.8 nautical miles against 60 at the equator, taking the earth as a sphere.

IMO resolution MSC.112(73) requires a GPS receiver to resolve 0.001 minute, in section 3.13, and to be accurate to within 100 m with 95% probability at an HDOP of 4, in section 3.4. Six decimals of a degree, about 0.11 m, state a position some 900 times finer than that.

Datum and chart shifts

Converting the form leaves the datum where it was. GPS positions are on WGS 84 under IMO resolution MSC.112(73), and ITU-R M.1371-6 has every AIS position sent on WGS 84. IHO S-4, B-201.2, makes WGS 84 the reference for nautical charts. B-201.3 still allows a local datum on paper charts, while electronic navigational charts must be on WGS 84.

A paper chart on another datum carries a SATELLITE-DERIVED POSITIONS note under IHO S-4, B-202, giving the shift in minutes north or south and east or west to apply to a GPS position before it is plotted. S-4, B-201.1, quotes IMO SN Circular 213: positions referred to different datums can differ by several hundred meters.

Where to find each input

A position in degrees and decimal minutes comes off the GPS or ECDIS display, the AIS target data or the deck log. Degrees, minutes and seconds appear on charts whose graduation is in seconds, which IHO S-4, B-131, allows, and in land survey and port documents. Decimal degrees come from GIS layers, web maps and spreadsheets, signed and with latitude first. A position from a DSC alert or a ship report under IMO resolution A.851(20) is in whole minutes: enter it as written, with no seconds.