Calculate the angle between two bearings and the shortest turn direction, or find the final bearing after a turn, with compass, quadrant and DMS results.
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Angle Between Bearings Formula
The main angle-between calculation compares two bearings after converting them to degrees and normalizing them to 0ยฐ inclusive through 360ยฐ exclusive. Use nonnegative modulo for the formulas below. The exterior angle E is a reference value, not a separate widget output; coincident bearings have a zero included angle, while 360 โ A describes a full revolution.
- 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:
For manual quadrant-bearing conversion before using the two-bearing mode, the azimuth depends on the starting direction, angle and ending direction. These are reference identities, not additional calculator tabs.
- theta = quadrant bearing angle, from 0ยฐ to 90ยฐ
- AZ = azimuth measured clockwise from north
The calculator reports the reciprocal final bearing. The following magnetic-declination identities are manual reference conversions; use a consistent north reference before comparing bearings:
- BAZ = back azimuth
- Declination = magnetic declination, with east positive and west negative
For manual conversion from planar coordinates before using this calculator, find the change in easting and northing, then use atan2 with east first and north second so that 0ยฐ points north. Coordinate entry is not a mode of this calculator.
- 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 reference, projecting a point in a plane from a start coordinate, distance and azimuth uses the following identities. This calculator does not take coordinate or projection inputs:
The two-bearing mode reports the smaller angle, both turn directions, a shortest-arc midpoint, normalized DMS bearings, nearest 16-point compass direction and reciprocal final bearing. The turn mode applies a clockwise or counterclockwise turn to a starting bearing, with a midpoint along the full entered turn. Equal bearings need no turn; opposite bearings have two equally short routes and two shortest-arc midpoints. Choose units beside each input and choose the primary result unit separately. DMS output rounds to 0.01 second with proper carry; DMS text is not accepted as input.
Bearing and Azimuth Reference Tables
Use these tables to check common direction conventions and interpret the calculator output.
| Direction | Azimuth | Quadrant bearing form |
|---|---|---|
| North | 0ยฐ or 360ยฐ | N 0ยฐ E |
| East | 90ยฐ | N 90ยฐ E |
| South | 180ยฐ | S 0ยฐ E |
| West | 270ยฐ | S 90ยฐ W |
| Azimuth range | Quadrant | Quadrant bearing angle |
|---|---|---|
| 0ยฐ to 90ยฐ | NE | N azimuth E |
| 90ยฐ to 180ยฐ | SE | S (180ยฐ - azimuth) E |
| 180ยฐ to 270ยฐ | SW | S (azimuth - 180ยฐ) W |
| 270ยฐ to 360ยฐ | NW | N (360ยฐ - azimuth) W |
Example Problems
Example 1: Find the smaller angle between two bearings
Suppose bearing 1 is 35ยฐ and bearing 2 is 120ยฐ.
The smaller angle between the bearings is 85ยฐ. The exterior angle is 275ยฐ.
Example 2: Find bearing from coordinates
Suppose point A is E1 = 100, N1 = 200 and point B is E2 = 160, N2 = 280.
The azimuth from A to B is about 36.869898ยฐ, which is N 36ยฐ 52โฒ 11.631โณ 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 0ยฐ to 360ยฐ. A quadrant bearing uses a north or south reference and an east or west direction, such as N 35ยฐ E or S 20ยฐ 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 0ยฐ, east equal 90ยฐ, south equal 180ยฐ, and west equal 270ยฐ.
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 80ยฐ and an east declination of 10ยฐ, the true bearing is 90ยฐ.
