Latitude and Longitude Coordinate System

The latitude and longitude coordinate system, used in GPS systems, is presented with examples. An online calculator to calculate the distance between two points on Earth is also included.

The Prime Meridian and the Equator

It is internationally agreed that the prime meridian is an imaginary great circle that divides the Earth (supposed to be a sphere) into two hemispheres—the eastern and western parts—as shown in Figures 1 and 2 below. The prime meridian circle runs through Greenwich, England.

The equator is also an imaginary circle that is equidistant from the North and South poles and divides the Earth into two hemispheres: the northern and southern parts.

Prime Meridian and the Equator
Fig.1 - The Prime Meridian and the Equator

The latitude is an angular measurement between the plane of the equator and the radius from the center of the Earth to the location considered. Therefore, locations on Earth with the same latitude form a circle called a line of latitude (see Figure 2).

The longitude is an angular measurement of the angle between the plane containing the prime meridian and the plane containing the North and South poles and the location considered. Points with the same longitude form a circle called a line of longitude or meridian.

Prime Meridian, Equator and latitude and longitude
Fig.2 - The Prime Meridian, the Equator, the Latitude, and the Longitude
View Example 1: Coordinates of Points A, B, C, and D

Example 1: Write the latitude and longitude of points A, B, C, and D (in Figure 2) using North (N), South (S), East (E), and West (W).

  1. Point A is at the intersection of the line with latitude \(30^\circ\) North and longitude \(60^\circ\) East. Written as: \(30^\circ\text{ N}, 60^\circ\text{ E}\)
  2. Point B is at latitude \(45^\circ\) N and longitude \(15^\circ\) W. Written as: \(45^\circ\text{ N}, 15^\circ\text{ W}\)
  3. Point C is at latitude \(15^\circ\) S and longitude \(75^\circ\) W. Written as: \(15^\circ\text{ S}, 75^\circ\text{ W}\)
  4. Point D is at latitude \(45^\circ\) S and longitude \(30^\circ\) E. Written as: \(45^\circ\text{ S}, 30^\circ\text{ E}\)

Range of Values of Latitude and Longitude

The latitude of the Equator is equal to \(0^\circ\). The longitude of the prime meridian (through Greenwich) is equal to \(0^\circ\).

The longitude takes values from \(0^\circ\) to \(180^\circ\) East and \(180^\circ\) West. Negative and positive signs may also be used to express longitude, such that longitude to the east is positive and longitude to the west is negative. The range of longitudes is written as: \([-180^\circ, 180^\circ]\).

The latitude takes values from \(0^\circ\) to \(90^\circ\) North and \(90^\circ\) South. Negative and positive signs may also be used, such that latitude to the North is positive and latitude to the South is negative. The range of latitudes is written as: \([-90^\circ, 90^\circ]\).

The whole earth surface
Fig.3 - The Whole World – Latitude \(90^\circ\) N to \(90^\circ\) S, Longitude \(180^\circ\) E to \(180^\circ\) W
View Example 2: Signed Coordinate Representations

Example 2: Write the coordinates using N/S, E/W and signed notation (omitting the positive \(+\) sign):

  • A: \(30^\circ\text{ N}, 60^\circ\text{ E}\) or \(30^\circ, 60^\circ\)
  • B: \(45^\circ\text{ N}, 15^\circ\text{ W}\) or \(45^\circ, -15^\circ\)
  • C: \(15^\circ\text{ S}, 75^\circ\text{ W}\) or \(-15^\circ, -75^\circ\)
  • D: \(45^\circ\text{ S}, 30^\circ\text{ E}\) or \(-45^\circ, 30^\circ\)

Examples of Latitude and Longitude of Capital Cities in Decimal Degrees

Below are examples using both signed/decimal notation and directional letters (S/N for latitude, E/W for longitude):

Formats in Writing Latitude and Longitude

Being angular measurements, latitude and longitude can be expressed in three standard formats:

  1. Decimal Degrees (DD): e.g., \(3.138675^\circ, 101.6169495^\circ\)
  2. Degrees, Minutes, and Seconds (DMS): e.g., \(3^\circ 34' 45.3'', 23^\circ 55' 23.7''\)
  3. Degrees and Decimal Minutes (DDM): e.g., \(32^\circ 34.67'\text{ S}, 43^\circ 55.67'\text{ W}\)

Convert Between Different Formats

Using conversion rates (\(1^\circ = 60'\) and \(1' = 60''\), meaning \(1'' = \frac{1}{3600}^\circ\)), conversions between DD, DMS, and DDM are performed as follows:

View Example 6: Format Conversion Steps

Example 6:

a) Convert \(3.138675^\circ\) to DDM and DMS:

\[ 3.138675^\circ = 3^\circ + (0.138675 \times 60)' = 3^\circ 8.3205' \quad (\text{DDM}) \] \[ = 3^\circ + 8' + (0.3205 \times 60)'' = 3^\circ 8' 19.23'' \quad (\text{DMS}) \]

b) Convert \(45^\circ 55' 23.7''\) to DD and DDM:

\[ 45^\circ 55' 23.7'' = 45^\circ + 55\left(\frac{1}{60}\right)^\circ + 23.7\left(\frac{1}{3600}\right)^\circ \approx 45.92325^\circ \quad (\text{DD}) \] \[ 45.92325^\circ = 45^\circ + (0.92325 \times 60)' = 45^\circ 55.395' \quad (\text{DDM}) \]

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