#### Calculating distances

The following example walks you through calculating the distances between two National Grid references.

#### Example locations

This example uses the following two points and calculates the distance between them:

- The former Ordnance Survey office in Southampton (Grid Referene = SU 387148)
- The Tower of London (Grid Reference = TQ 336805),

See the steps below to make the calculation:

1) Use the diagram below to replace the grid letters with their corresponding numbers.

>> IMAGE << See the example in pdf format below.

Example of distance calculations

#### Changing the references:

- For Ordnance Survey, SU becomes 41, giving a revised reference of
**41**/387148 - For the Tower of London, TQ becomes 51, giving
**51**/336805

(2) Split each set of numerals in half to give Eastings and Northings. Thus 41/387148 becomes E 4387 and N 1148, and 51/336805 becomes E 5336 and N 1805.

The number of digits in each Easting or Northing set indicates the precision. If three digits, the result would be in kilometres, four digits would give a result in units of 100 m, five digits a result in units of 10 m, and six digits a result in metres. Thus 4387, being four digits, would give an answer in units of 100 metres.

(3) Pythagoras' theorem states that in a right angled triangle, the square of the length of the hypotenuse equals the sum of the squares of the other two sides. As Eastings lines are perpendicular with Northings, we can use the theorem to calculate the distance between the two points (the hypotenuse).

>> IMAGE: Triangulation example <<

(4) Calculate the difference in Eastings by subtracting the smaller figure from the larger, eg 5336 minus 4387 gives 949. Calculate the difference in Northings by subtracting the smaller figure from the larger, eg 1805 minus 1148 gives 657.

(5) Square the two results and add together,

949^{2} + 657^{2} = 1 332 250

The square root of this result is the length of the hypotenuse,

1 332 250 = 1154

(6) The distance between the two points is therefore 1154 units, and as 1 unit in this example represents 100 m (see para 3 above) the distance becomes 115 400m or 115.4 km.

(7) Thus the distance between Ordnance Survey and The Tower of London is 115.4 km.

**Please note: **this is for two points on a projection (plane surface) and does not take into account either the Earth's curvature or local or line scale factor.