Calculate the angle between two bearings or azimuths, convert quadrant bearings and find bearings from coordinates or projected points.

Angle Between Bearings Calculator

Choose a tab, enter values, then calculate.

Angle Between
Bearing Converter
From Coordinates
Project Point
Quadrant bearing input
Azimuth or compass input
Show Calculation Steps ▼

Related Calculators

Angle Between Bearings Formula

The main angle-between calculation compares two bearings or azimuths after converting them to degrees and normalizing them to the range 0° to 360°.

CW = (B2 - B1) mod 360
CCW = (B1 - B2) mod 360
A = min(CW, CCW)
E = 360 - A
  • B1 = first bearing or azimuth, in degrees after conversion if needed
  • B2 = second bearing or azimuth, in degrees after conversion if needed
  • CW = clockwise angle from bearing 1 to bearing 2
  • CCW = counterclockwise angle from bearing 1 to bearing 2
  • A = smaller angle between the bearings
  • E = exterior angle

For radians, the calculator uses this conversion before or after the bearing calculation:

deg = rad*180/pi
rad = deg*pi/180

For quadrant bearing conversion, the azimuth depends on the starting direction, the angle, and the ending direction.

N theta E: AZ = theta
S theta E: AZ = 180 - theta
S theta W: AZ = 180 + theta
N theta W: AZ = 360 - theta
  • theta = quadrant bearing angle, from 0° to 90°
  • AZ = azimuth measured clockwise from north

For back azimuths and magnetic declination conversions, the calculator uses:

BAZ = (AZ + 180) mod 360
True = Magnetic + Declination
Magnetic = True - Declination
  • BAZ = back azimuth
  • Declination = magnetic declination, with east positive and west negative

For a bearing from coordinates, the calculator finds the change in easting and northing, then uses atan2 with east first and north second so that 0° points north.

dE = E2 - E1
dN = N2 - N1
AZ = atan2(dE, dN)*180/pi
D = sqrt(dE^2 + dN^2)
  • E1, N1 = starting easting/X and northing/Y
  • E2, N2 = ending easting/X and northing/Y
  • dE = change in easting or X
  • dN = change in northing or Y
  • D = straight-line distance between the two points

For projecting a point from a start coordinate, distance, and azimuth, the calculator uses:

dE = D*sin(AZ)
dN = D*cos(AZ)
E2 = E1 + dE
N2 = N1 + dN

The angle-between tab returns the smaller angle, clockwise angle, counterclockwise angle, exterior angle, line angle, and smaller-angle bisector. The converter tab changes between azimuths, quadrant bearings, back bearings, and magnetic or true bearings. The coordinate tab calculates bearing and distance from two points. The project point tab calculates the endpoint from a starting point, distance, and azimuth.

Bearing and Azimuth Reference Tables

Use these tables to check common direction conventions and interpret the calculator output.

Direction Azimuth Quadrant bearing form
North 0° or 360° N 0° E
East 90° N 90° E
South 180° S 0° E
West 270° S 90° W
Azimuth range Quadrant Quadrant bearing angle
0° to 90° NE N azimuth E
90° to 180° SE S (180° - azimuth) E
180° to 270° SW S (azimuth - 180°) W
270° to 360° NW N (360° - azimuth) W

Example Problems

Example 1: Find the smaller angle between two bearings

Suppose bearing 1 is 35° and bearing 2 is 120°.

CW = (120 - 35) mod 360 = 85
CCW = (35 - 120) mod 360 = 275
A = min(85, 275) = 85

The smaller angle between the bearings is 85°. The exterior angle is 275°.

Example 2: Find bearing from coordinates

Suppose point A is E1 = 100, N1 = 200 and point B is E2 = 160, N2 = 280.

dE = 160 - 100 = 60
dN = 280 - 200 = 80
AZ = atan2(60, 80)*180/pi = 36.869898
D = sqrt(60^2 + 80^2) = 100

The azimuth from A to B is about 36.869898°, which is N 36° 52′ 11.631″ E. The distance is 100 units.

FAQ

What is the difference between a bearing and an azimuth?

An azimuth is a single angle measured clockwise from north, usually from 0° to 360°. A quadrant bearing uses a north or south reference and an east or west direction, such as N 35° E or S 20° W. Both describe direction, but the notation is different.

Why does the coordinate formula use atan2(dE, dN) instead of atan2(dN, dE)?

Bearings and azimuths are measured clockwise from north. In coordinate geometry, the usual x-axis angle is measured from east, but a bearing starts from north. Using atan2(dE, dN) makes north equal 0°, east equal 90°, south equal 180°, and west equal 270°.

How do you handle magnetic declination?

Use east declination as positive and west declination as negative. To convert magnetic to true, add declination. To convert true to magnetic, subtract declination. For example, with a magnetic bearing of 80° and an east declination of 10°, the true bearing is 90°.