# 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.

## Precision

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 111.32 km. As one moves away from the equator towards a pole, however, one degree of longitude is multiplied by the cosine of the latitude, decreasing the distance, approaching zero at the pole. The number of decimal places required for a particular precision at the equator is:

Decimal degree precision versus length
decimal
places
degrees N/S or
E/W at equator
E/W at
23N/S
E/W at
45N/S
E/W at
67N/S
0 1.0 111.32 km 102.47 km 78.71 km 43.496 km
1 0.1 11.132 km 10.247 km 7.871 km 4.3496 km
2 0.01 1.1132 km 1.0247 km .7871 km 0.43496 km
3 0.001 111.32 m 102.47 m 78.71 m 43.496 m
4 0.0001 11.132 m 10.247 m 7.871 m 4.3496 m
5 0.00001 1.1132 m 1.0247 m .7871 m 0.43496 m
6 0.000001 111.32 mm 102.47 mm 78.71 mm 43.496 mm
7 0.0000001 11.132 mm 10.247 mm 7.871 mm 4.3496 mm
8 0.00000001 1.1132 mm 1.0247 mm .7871 mm 0.43496 mm

A value in decimal degrees to an precision of 4 decimal places is precise to 11.132 meters at the equator. A value in decimal degrees to 5 decimal places is precise to 1.1132 meter at the equator. Elevation also introduces a small error. At 6,378 m elevation, the radius and surface distance is increased by 0.001 or 0.1%. Because the earth is not flat, the precision of the longitude part of the coordinates increases the further from the equator you get. The precision 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

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

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

For instance, the decimal degree representation for

38° 53′ 23″ N, 77° 00′ 32″ W

(the location of the United States Capitol) is

38.889722°, -77.008889°

In most systems, such as Google Maps, the degree symbols are omitted, reducing the representation to

38.889722, -77.008889

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

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

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.

Pseudo code, using numbers from above:
$min=int(mod(38.889722*60,60));$sec=round(mod(38.889722*3600,60)) => min:53, sec:23