Question 1: Bearing and Distance From Coordinates
Two points have the following coordinates:
Point
A
B
Easting Northing
32255.751 49076.286
12231.864 36939.667
Calculate the bearing A to B and plan distance AB.
Answer: 238°46’46.7”, 23414.815 m
Solution:
Visualise the problem.
55000
N
Northing
A
45000
N
B
35000
10000
E
20000
30000
Easting
© 2002 School of Surveying and Spatial Information Systems, UNSW
40000
Now find the E and N:
Note: To find the bearing
of A to B we take B
coordinates minus A
coordinates
E = EB - EA
= 12231.864 - 32255.751
= -20023.887
N = NB - NA
= 36939.667 - 49076.286
= -12136.619
Using Pythagoras’s Theorem to solve dAB:
d AB     
2
55000
2
= -20023.887 + -12136.619
2
N
A
2
Northing
d AB  - 20023.887  - 12136.619
2
45000
N
dAB = 23414.815 m
B
Now for the Bearing :
35000
10000
E
20000
30000
Easting
 E 
  Tan 1 

 N 
The signs of E and N will determine the quadrant of the bearing:
+ N
0°
- E
270°
B
A
90°
+ E
180°
- N
The diagram above shows that our bearing will be in the 3rd quadrant, between 180°
and 270°
© 2002 School of Surveying and Spatial Information Systems, UNSW
40000
 - 20023.887 
  Tan 1 

 - 12136.619 
= 58.77965125
This is in the wrong quadrant so we have to add 180°
58°.77965125 + 180°
55000
N
= 238.77965125
The degrees is just the integer of 238.77965125:
Northing
A
To convert to degrees, minutes and seconds:
45000
N
Degrees = 238º
B
Minutes = (238.77965125 - 238)  60
= 46.779075 (take the integer of this)
= 46’
Seconds = (238.77965125 - 238 -
35000
10000
E
20000
30000
Easting
46
)  3600
60
= 46.7”
Therefore:
 = 238°46’46.7”
Check by reverse solution:
We can check our result by working backwards starting at point A and calculating the
coordinates of B using the derived bearing and distance.
ΔE AB  d AB sin β AB
= 23414.815  sin(238°46’46.7”)
= -20023.884
ΔN AB  d AB cos β AB
= 23414.815  cos(238°46’46.7”)
= -12136.623
Coordinates of B:
EB = 32255.751 + -20023.884
© 2002 School of Surveying and Spatial Information Systems, UNSW
40000
= 12231.867
NB = 49076.286 + -12136.623
= 36939.663
Now you will notice that the easting and northing of B differs by 3mm and 4mm
respectively. This is due to the rounding of our bearing (0.1” error over our 23km line
represents about 11mm).
l=r
= 24000  0.1”  
= 11mm
11mm
0.1”
A
B
24 km
Excel Example:
55000
N
Northing
A
45000
N
B
35000
10000
E
20000
30000
Easting
Formulae:
© 2002 School of Surveying and Spatial Information Systems, UNSW
40000
Now try solving this question yourself.
Two points have the following coordinates:
Point
A
B
Easting Northing
32.751 50076.286
121.864 49939.667
Calculate the bearing A to B and distance AB.
(Check your answer using the above methods).
© 2002 School of Surveying and Spatial Information Systems, UNSW