Decimal degrees

Decimal degrees (DD) express latitude and longitude geographic coordinates as decimal fractions and are used in many Geographic Information Systems (GIS), web mapping applications such as Google Maps, and GPS devices. Decimal degrees are an alternative to using degrees, minutes, and seconds (DMS). As with latitude and longitude, the values are bounded by ±90° and ±180° respectively.

Positive latitudes are north of the equator, negative latitudes are south of the equator. Positive longitudes are east of Prime Meridian, negative longitudes are west of the Prime Meridian. Latitude and longitude are usually expressed in that sequence, latitude before longitude.

Accuracy

The radius of the semi major axis of the Earth at the equator is 6,378,160.0 meters^[1] resulting in a circumference of 40,075,161.2 meters. The equator is divided into 360 degrees of longitude, so each degree at the equator represents 111,319.9 metres or approximately 111 km. As one moves away from the equator towards a pole, however, one degree of longitude represents a diminishing number of meters, approaching zero at the pole. The number of decimal places required for a particular accuracy at the equator is:

Accuracy versus decimal places at the equator

decimal places	degrees	distance
0	1.0	111 km
1	0.1	11.1 km
2	0.01	1.11 km
3	0.001	111 m
4	0.0001	11.1 m
5	0.00001	1.11 m
6	0.000001	0.111 m
7	0.0000001	1.11 cm
8	0.00000001	1.11 mm

A value in decimal degrees to an accuracy of 4 decimal places is accurate to 11.1 meters (± 5.55 m) at the equator. A value in decimal degrees to 5 decimal places is accurate to 1.11 meter at the equator. Because the earth is not flat, the accuracy of the longitude part of the coordinates increases the further from the equator you get. The accuracy of the latitude part does not increase so much, more strictly however, a meridian arc length per 1 second depends on latitude at point concerned. The discrepancy of 1 second meridian arc length between equator and pole is about 0.3 metres because the earth is an oblate spheroid.

Example

The decimal degree representation of the location of the United States Capitol is:

38.889722°, -77.008889°

In most systems, such as Google Maps, the degree symbol is omitted and thus the coordinates are expressed as "38.889722, -77.008889".

A DMS value is converted to decimal degrees using the formula:

$\mathrm{DD} = \mathrm{D} + \frac{\mathrm{M}}{60} + \frac{\mathrm{S}}{3600}$

The decimal degree representation for the location of the United States Capitol of

38° 53′ 23″ N, 77° 00′ 32″ W

is:

38.889722°, -77.008889°

To calculate the D, M and S components, the following formulas can be used:

$\mathrm{DD} = \begin{cases} \mathrm{D} &= \operatorname{trunc}(\mathrm{DD}) \\ \mathrm{M} &= \operatorname{trunc}(|\mathrm{DD}| \cdot 60) \bmod 60 \\ \mathrm{S} &= \left(|\mathrm{DD}| \cdot 3600 \right) \bmod 60 \end{cases}$

where |DD| is the absolute value of DD, trunc is the truncation function, and mod is the modulo operator. Note that with this formula, only S may have a fractional value.

References

1. ^ Australian Geodetic Datum