Latitude and longitude are the coordinates of the ellipsoid itself, before any projection is applied. Everything else in this section — State Plane, UTM, low-distortion projections — is a transformation of these two angles into a plane grid. They are the underlying quantity, and they are also the format in which coordinates are most often mangled.
The mangling is almost never mathematical. It is notational: a missing sign, a hemisphere letter dropped, decimal minutes read as decimal degrees. Each of those moves a point by anything from a few hundred meters to the other side of the world.
The two angles
- Latitude (φ)
- The angle between the equatorial plane and the normal to the ellipsoid at the point, measured north or south from 0° at the equator to 90° at the poles. On an ellipsoid this is geodetic latitude, which differs slightly from geocentric latitude — the angle subtended at the center of the ellipsoid — by up to about 11.5′ at mid-latitudes.
- Longitude (λ)
- The angle in the equatorial plane between the prime meridian and the meridian through the point, measured 0° to 180° east or west. Unlike latitude, longitude has no natural origin — the prime meridian is a convention.
The geodetic versus geocentric distinction is worth knowing about even though it rarely arises in practice. All published survey latitudes are geodetic, and it is only in satellite and gravity work that the geocentric form appears. If nothing states otherwise, geodetic is meant.
How far is a degree?
| Increment | Latitude, anywhere | Longitude at the equator | Longitude at 40° N |
|---|---|---|---|
| 1° | ≈ 111 km | ≈ 111 km | ≈ 85.3 km |
| 1′ | ≈ 1853 m | ≈ 1855 m | ≈ 1421 m |
| 1″ | ≈ 30.9 m | ≈ 30.9 m | ≈ 23.7 m |
| 0.1″ | ≈ 3.1 m | ≈ 3.1 m | ≈ 2.4 m |
| 0.001″ | ≈ 31 mm | ≈ 31 mm | ≈ 24 mm |
The asymmetry is the key fact. A degree of latitude is essentially the same length everywhere, because meridians are all much the same. A degree of longitude shrinks by the cosine of the latitude, from about 111 km at the equator to about 85.3 km at 40° north and to zero at the pole.
The practical consequence is that latitude and longitude need different numbers of decimal places to express the same ground precision, and that a coordinate quoted to a fixed number of decimals is more precise in longitude at high latitudes than the digits suggest.
The three notations
| Format | Example | Where it is used |
|---|---|---|
| Degrees, minutes, seconds (DMS) | 39° 44′ 21.6″ N, 104° 59′ 04.9″ W | Surveying, published control, legal descriptions |
| Degrees and decimal minutes (DDM) | 39° 44.3600′ N, 104° 59.0817′ W | Marine and aeronautical navigation, GPS handhelds |
| Decimal degrees (DD) | 39.7393333, −104.9847 | GIS, databases, web mapping, computation |
Conversion between them is elementary — divide the seconds by 3600 and the minutes by 60 and add — but the direction of the sign convention is not, and the mixed forms are where errors live. Degrees and decimal minutes is particularly hazardous in text, because 39° 44.36′ and 39.4436° look similar and differ by about 33 km.
decimal degrees = degrees + minutes ÷ 60 + seconds ÷ 3600Signs and hemispheres
The safe practice is to pick one convention and enforce it. Either use signed decimal degrees throughout, with west and south negative, or use unsigned magnitudes with an explicit hemisphere letter that is never separated from its number. Mixing the two — a negative number and a W — is how a longitude ends up doubly negated.
Precision, and being honest about it
Five decimal places of latitude is about 1.1 m. Six is about 0.11 m. Seven is about 11 mm. Publishing a latitude to seven decimals from a handheld receiver claims millimeter knowledge you do not have, and the claim will be believed by whoever inherits the file.
In DMS, seconds to two decimal places is about 0.3 m of latitude, and to three decimal places about 31 mm. Geodetic control is normally published to five decimal places of a second, which corresponds to a fraction of a millimeter — not because the position is that good, but so that no precision is lost in the last conversion. Both the value and the stated uncertainty should travel together.
The part that gets forgotten
A latitude and longitude are meaningless without a datum. The same physical monument has different geographic coordinates on NAD 27, on NAD 83 (2011) and on a contemporary global frame — differing by tens of meters in the first comparison and by one to two meters in the second. Unlike a State Plane coordinate, where a change of datum shifts the numbers so dramatically that nobody could mistake them, a NAD 27 latitude and a NAD 83 latitude look equally plausible.
- Always record the datum and realization with a geographic coordinate. This matters more here than anywhere else in coordinate work.
- Choose one notation for a dataset and stay in it.
- Use signed decimal degrees for storage and computation; convert to DMS only for presentation.
- Use the typographic prime and double prime for minutes and seconds, not the apostrophe and quotation mark, so the values survive parsing.
- State precision honestly — the number of decimals should reflect the actual accuracy, not the display setting.
- Check the sign of a western longitude before anything else. It is the highest-consequence single character in the coordinate.