Degrees, minutes and seconds are a base-60 way of writing an angle, inherited from Babylonian astronomy and still in daily use on survey plans, nautical charts and aviation documents. The arithmetic is simple — 60 minutes to a degree, 60 seconds to a minute — but the format carries a hidden trap: the number of decimal places does not tell you how precise the angle is, and two angles written to different precisions are not the same measurement. This guide covers the conversions and the precision question.
| Unit | Symbol | Definition | Notation |
|---|---|---|---|
| Degree | ° | 1/360 of a full circle | 40° |
| Minute | ′ | 1/60 of a degree | 45′ |
| Second | ″ | 1/60 of a minute, 1/3600 of a degree | 55.56″ |
A single angle written in full looks like 40° 45′ 55.56″ N. The hemisphere letter is part
of the notation, not decoration: without it, the sign of the angle has to be carried as a minus for
south and west, which is how decimal degrees handle it.
Decimal degrees is the format software prefers, because it is a single number that can be sorted, interpolated and used in trigonometry directly. The conversion is:
Take 40° 45′ 55.56″. The minutes contribute 45 ÷ 60 = 0.75, and the seconds
contribute 55.56 ÷ 3600 = 0.01543. Adding them to the 40° gives
40.76543°. A southern or western angle carries a minus sign, so
104° 59′ 15.56″ W becomes −104.98766°.
Going the other way reverses the steps, and it is an iterative process rather than one formula:
For 38.8977°: the degrees are 38, the fraction 0.8977 × 60 = 53.862 minutes, and the
remaining 0.862 × 60 = 51.72 seconds. So 38.8977° is
38° 53′ 51.72″. There is one subtlety worth knowing: rounding can produce
60″, which is invalid and has to be carried into the minutes, and then possibly into the
degrees. Any converter that can output 60″ has a bug that only appears on specific
inputs.
Degrees and decimal minutes — DDM — is a hybrid, written as 40° 45.926′ N. It is common
in marine navigation because a minute of latitude is a nautical mile, so decimal minutes map directly
onto distance. DDM is neither DMS nor decimal degrees, and mixing it into a calculator that expects
one of the other two produces an error that is small enough to miss: reading
45.926′ as 45.926 seconds instead of minutes puts the point about 1.4 km away.
Latitude changes by about 111.32 km per degree, so the value of a unit is easy to work out. The same calculation for longitude depends on latitude, because lines of longitude converge towards the poles.
| Angular units | Latitude | Longitude at 45° N/S | Type of work it supports |
|---|---|---|---|
| 1° | 111.3 km | 78.7 km | Regional |
| 1′ | 1,855 m | 1,312 m | Map sheet |
| 1″ | 30.9 m | 21.9 m | Field navigation |
| 0.1″ | 3.1 m | 2.2 m | Handheld GNSS |
| 0.01″ | 0.31 m | 0.22 m | Mapping-grade GNSS |
| 0.001″ | 3.1 cm | 2.2 cm | Survey-grade GNSS |
| 0.0001° | 11.1 m | 7.9 m | Six decimal places of DD |
The table explains why survey records so often give seconds to two decimal places: that is 0.31 m, which is the right order for legal boundary description and comfortably finer than most total station work. It also explains why giving latitude and longitude to eight decimal places in decimal degrees is not more accurate — it is the same measurement written with more digits, and the extra digits are noise.
Trailing zeros are meaningful here. In DMS notation, 45.10″ and
45.1″ denote the same angle but claim different precision — one is stated to a
hundredth of a second, the other to a tenth. When a record rounds, report the rounding rather than
silently extending the digits.
Decimal degrees dominates GIS, web mapping and any dataset that a computer will process. DMS dominates survey plats, deeds, aviation and legacy map annotation, because a person reads it and because a minute of latitude is a known distance. DDM dominates marine navigation. The practical consequence is the same in every case: a coordinate without a stated format is ambiguous, and the ambiguity is only visible on the rare input where the two readings differ enough to notice.
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