How to Calculate Azimuth

Azimuth is the direction of a line measured clockwise from north.

It is usually written as an angle from 0° to 360°.

North = 0° or 360°
East  = 90°
South = 180°
West  = 270°

For example, an azimuth of 90° points east, while an azimuth of 225° points southwest.

Azimuth is used in navigation, surveying, mapping, astronomy, antenna alignment, hiking, aviation, and coordinate calculations.

What Is Azimuth?

Azimuth describes direction as a full-circle angle.

The reference direction is usually true north, and the angle increases clockwise.

          0° / 360°
             N
             |
270° W ------+------ E 90°
             |
             S
            180°

So if you are standing at one point and looking toward another point, the azimuth tells you the direction to face.

An azimuth answers:

What direction is the target from my starting point?

Azimuth Formula for Coordinates

If you know two latitude and longitude points, you can calculate the initial azimuth from the first point to the second point.

Use:

θ = atan2(sin(Δλ) × cos(φ₂), cos(φ₁) × sin(φ₂) - sin(φ₁) × cos(φ₂) × cos(Δλ))

Where:

θ = initial azimuth
φ₁ = latitude of the starting point
φ₂ = latitude of the destination point
λ₁ = longitude of the starting point
λ₂ = longitude of the destination point
Δλ = λ₂ - λ₁

Latitudes and longitudes should be converted to radians before using trigonometric functions.

After calculating θ, convert it from radians to degrees and normalize it to a value from 0° to 360°.

Use:

Azimuth = (θ × 180/π + 360) mod 360

This keeps negative angles from appearing in the final result.

Step-by-Step Azimuth Calculation

To calculate azimuth between two coordinate points:

Convert both latitudes from degrees to radians.
Convert both longitudes from degrees to radians.
Find the difference in longitude.
Use the atan2 azimuth formula.
Convert the result from radians to degrees.
Normalize the angle to 0° through 360°.

The coordinate formula is more complex than a simple flat-map angle because Earth is curved.

For short distances, a plane approximation may be close enough. For geographic coordinates, the spherical formula is usually more appropriate.

Example: Simple Compass Azimuth

If a direction points exactly east, the azimuth is:

90°

If a direction points exactly south, the azimuth is:

180°

If a direction points exactly northwest, the azimuth is halfway between west and north:

315°

These compass directions are useful for understanding azimuth before working with coordinates.

Compass Direction to Azimuth Chart

Common compass directions can be written as azimuth angles.

Direction	Azimuth
North	0° or 360°
Northeast	45°
East	90°
Southeast	135°
South	180°
Southwest	225°
West	270°
Northwest	315°

For more precision, compass directions can be split into smaller increments.

For example:

NNE = 22.5°
ENE = 67.5°
ESE = 112.5°
SSW = 202.5°
WNW = 292.5°

Azimuth vs Bearing

Azimuth and bearing both describe direction, but they are commonly written differently.

Azimuth uses a full circle from 0° to 360°.

Bearing often uses a quadrant format such as:

N 45° E
S 30° W

For example:

Azimuth	Quadrant Bearing
45°	N 45° E
120°	S 60° E
210°	S 30° W
300°	N 60° W

Azimuth is usually easier for calculators and coordinate work because it uses one continuous angle scale.

Quadrant bearings are common in surveying and some navigation contexts.

How to Convert Bearing to Azimuth

For quadrant bearings:

Bearing Format	Azimuth Rule
N θ E	`θ`
S θ E	`180° - θ`
S θ W	`180° + θ`
N θ W	`360° - θ`

Example: Convert S 40° E

Use:

180° - 40° = 140°

So:

S 40° E = 140° azimuth

Example: Convert N 25° W

Use:

360° - 25° = 335°

So:

N 25° W = 335° azimuth

How to Convert Azimuth to Bearing

To convert azimuth to quadrant bearing, first identify the quadrant.

Azimuth Range	Bearing Format
0° to 90°	N θ E
90° to 180°	S `(180° - azimuth)` E
180° to 270°	S `(azimuth - 180°)` W
270° to 360°	N `(360° - azimuth)` W

Example: Convert 240° Azimuth

The angle is between 180° and 270°, so it is in the southwest quadrant.

Use:

240° - 180° = 60°

So:

240° azimuth = S 60° W

Example: Convert 310° Azimuth

The angle is between 270° and 360°, so it is in the northwest quadrant.

Use:

360° - 310° = 50°

So:

310° azimuth = N 50° W

True North vs Magnetic North

Azimuth is often measured from true north, but a physical compass points toward magnetic north.

These are not always the same.

The difference between true north and magnetic north is called magnetic declination.

If you are using a compass, map, or GPS, check whether your direction is based on:

true north
magnetic north
grid north

This matters because a few degrees of difference can become important over long distances.

For short casual navigation, the difference may not matter much. For surveying, aviation, marine navigation, and long-distance travel, it can matter a lot.

Forward Azimuth and Back Azimuth

The forward azimuth is the direction from the starting point to the destination.

The back azimuth is the opposite direction, from the destination back to the starting point.

To calculate a simple back azimuth:

If the azimuth is less than 180°, add 180°.
If the azimuth is greater than or equal to 180°, subtract 180°.

Example: Forward Azimuth 70°

Since 70° is less than 180°:

70° + 180° = 250°

Back azimuth:

250°

Example: Forward Azimuth 230°

Since 230° is greater than 180°:

230° - 180° = 50°

Back azimuth:

50°

For long geographic paths, the exact reverse direction may vary slightly because great-circle routes curve on the globe.

Azimuth on a Flat Map

If you are working on a flat coordinate grid, azimuth can be calculated from coordinate differences.

Let:

ΔE = change in east-west coordinate
ΔN = change in north-south coordinate

Then:

Azimuth = atan2(ΔE, ΔN)

This differs from the usual math angle formula because azimuth is measured clockwise from north, not counterclockwise from the positive x-axis.

After converting to degrees:

Azimuth = (angle + 360) mod 360

This method is useful for grid coordinates, local maps, and surveying calculations where curvature is not important.

Example: Flat Grid Azimuth

Suppose a point moves:

ΔE = 30
ΔN = 40

The direction is east and north, so the azimuth should be between 0° and 90°.

Use:

Azimuth = atan2(30, 40)

This gives approximately:

36.87°

So the azimuth is:

36.87°

The line points northeast, closer to north than east.

Where Azimuth Is Used

Azimuth appears in many practical fields.

Pilots, sailors, hikers, and map users use azimuth to describe travel direction.

For example:

Follow an azimuth of 75°

means travel roughly east-northeast.

Surveying

Surveyors use azimuth and bearings to describe property lines, boundaries, and measured directions.

Mapping and GIS

Geographic information systems use azimuth to calculate direction between points, line orientation, and spatial relationships.

Astronomy

In astronomy, azimuth describes the horizontal direction of an object in the sky.

It is often paired with altitude, which describes how high the object is above the horizon.

Solar Panels and Antennas

Azimuth helps describe the horizontal direction a solar panel, satellite dish, or antenna faces.

Common Azimuth Mistakes

Measuring From East Instead of North

In standard math, angles are often measured from the positive x-axis.

Azimuth is measured from north.

That difference changes the formula and the interpretation.

Measuring Counterclockwise

Azimuth increases clockwise:

North = 0°
East = 90°
South = 180°
West = 270°

If you measure counterclockwise, the direction will be wrong.

Forgetting to Normalize the Angle

Some formulas return negative angles.

For example:

-45°

As an azimuth, this should be written as:

315°

Use:

(angle + 360) mod 360

Confusing True and Magnetic North

A coordinate formula usually gives a true azimuth.

A compass may show magnetic direction.

If precision matters, account for magnetic declination.

Assuming Back Azimuth Is Always Exact on the Globe

For simple compass work, adding or subtracting 180° is usually enough.

For long-distance great-circle paths, the initial azimuth from point A to point B is not always exactly opposite the initial azimuth from point B to point A.

Quick Azimuth Reference

Direction	Azimuth
N	0° or 360°
NE	45°
E	90°
SE	135°
S	180°
SW	225°
W	270°
NW	315°

Useful formulas:

Task	Formula
Coordinate azimuth	`atan2(sin(Δλ) × cos(φ₂), cos(φ₁) × sin(φ₂) - sin(φ₁) × cos(φ₂) × cos(Δλ))`
Flat grid azimuth	`atan2(ΔE, ΔN)`
Normalize angle	`(angle + 360) mod 360`
Back azimuth under 180°	`azimuth + 180°`
Back azimuth 180° or more	`azimuth - 180°`

Frequently Asked Questions

What is azimuth?

Azimuth is a direction angle measured clockwise from north.

It is usually written from 0° to 360°.

What is an azimuth of 90°?

An azimuth of 90° points east.

What is an azimuth of 180°?

An azimuth of 180° points south.

What is an azimuth of 270°?

An azimuth of 270° points west.

Is azimuth the same as bearing?

Not always.

Azimuth is usually written as one angle from 0° to 360°.

Bearing may be written in quadrant form, such as N 45° E or S 30° W.

How do you calculate back azimuth?

If the azimuth is less than 180°, add 180°.

If the azimuth is 180° or more, subtract 180°.

For example:

70° + 180° = 250°

So the back azimuth of 70° is 250°.

What is the difference between true azimuth and magnetic azimuth?

True azimuth is measured from true north.

Magnetic azimuth is measured from magnetic north.

The difference between them is magnetic declination.

Can azimuth be negative?

Azimuth is normally written from 0° to 360°.

If a formula gives a negative angle, add 360°.

For example:

-30° + 360° = 330°

So -30° is written as 330° azimuth.

For coordinate-based direction work, use the Azimuth Calculator. For an external reference on azimuth as a 360-degree compass direction, see the National Weather Service azimuth glossary entry; for true vs magnetic north, NOAA’s magnetic declination page is helpful.

Final Thoughts

Azimuth is a compact way to describe direction.

The key idea is simple:

Azimuth is measured clockwise from north.

North is 0°, east is 90°, south is 180°, and west is 270°.

For compass directions, this is straightforward. For latitude and longitude, use the coordinate azimuth formula and normalize the result to 0° through 360°.

Before using an azimuth in the real world, check whether it is based on true north, magnetic north, or grid north. That small detail can matter more than the calculation itself.