Azimuth Calculator - Bearing, Quadrant, and Great-Circle Distance

An azimuth calculator returns the initial bearing between any two points on Earth when you give it the latitude and longitude of each point, along with the great-circle distance and a compass quadrant label.

• • Satellite dish alignment: Point a dish at a target geostationary satellite and read the bearing in degrees clockwise from true north.
• • Solar panel orientation planning: Check the roof azimuth relative to true south and pick the equator-facing slope that maximizes annual generation.
• • Hiking and land navigation: Turn two map coordinates into a single compass bearing for a handheld compass when the trail fades.
• • Drone and aircraft course planning: Set the initial heading for a UAV or light-aircraft leg between two waypoints, then compare the bearing to the actual track to spot wind drift.

Azimuth has been the standard horizontal direction in navigation for centuries. The U.S. Army Field Manual 21-26 defines azimuth as a horizontal angle measured clockwise from a north base line, with east at 90 degrees, south at 180 degrees, and west at 270 degrees.

This page computes the initial bearing and great-circle distance in one pass. A London-to-Rio leg comes out close to 9,300 km even though the straight line on a flat map looks longer.

When you want the point that lies in the exact opposite direction from the starting coordinate, the Antipode Calculator returns the diametrically opposite latitude and longitude in the same signed-decimal format.

The azimuth calculator converts each latitude and longitude from signed decimal degrees to radians, then plugs them into the standard initial-bearing equation and the haversine distance equation. Both come from spherical trigonometry and assume Earth is a sphere with the mean radius used by NASA.

• phi_1, phi_2: Latitude of the starting and destination points, in radians.
• Delta_lambda: Longitude difference between destination and starting points, in radians.
• atan2(y, x): Two-argument arctangent that returns the angle in the correct quadrant from the east-west numerator and north-south denominator.
• R = 6371.0088 km: Earth's mean radius, the multiplier that turns the central angle from radians into kilometers.

atan2 is used instead of plain atan because atan only returns a value between -90 and +90 degrees, which collapses the compass onto a single line. atan2 takes the east-west and north-south components separately and returns the angle in the correct quadrant.

The haversine formula runs on the same coordinates to get the distance, using the half-angle sine identity to avoid the cancellation error you get from the spherical law of cosines when two points are close together.

According to Wikipedia (Azimuth), azimuth is the horizontal angle measured clockwise from a north base line in a local spherical coordinate system, with east at 90 degrees, south at 180 degrees, and west at 270 degrees.

If you only need the distance along the great-circle arc and not the bearing, Great Circle Calculator solves the haversine half of this calculation in a single keystroke.

Four ideas make the formula's output interpretable: what an azimuth is, how atan2 fixes the quadrant, why Earth is treated as a sphere, and how compass quadrants map to the numeric bearing.

Once you know the azimuth comes from atan2, the same number describes the compass heading for a telescope, dish, rifle scope, or theodolite without any further conversion.

If the two points are exact antipodes, the initial bearing is mathematically undefined, so the page returns about 20,015 km and a bearing of 0.00 degrees to keep the result table readable.

The same idea of measuring horizontal angle from a reference axis shows up in physics, and Spherical Coordinates Calculator walks through the related conversion for spherical coordinates in three dimensions.

Five short steps take you from two coordinates on a map to the initial bearing, compass quadrant, and great-circle distance.

1. 1 Look up the starting coordinates: Copy the starting latitude and longitude from a map app. The defaults are central London.
2. 2 Look up the destination coordinates: Copy the destination latitude and longitude in the same signed decimal-degree format. The default is central Rio de Janeiro.
3. 3 Enter the starting latitude and longitude: Type the starting latitude in the first row and the longitude next to it. Use the sign convention of the source: positive north and east, negative south and west.
4. 4 Enter the destination latitude and longitude: Type the destination latitude and longitude in the second row. The page validates each value and stops with a clear error if a value is out of bounds.
5. 5 Read the azimuth, quadrant, and distance: The initial bearing is shown in degrees clockwise from true north, the 8-point compass quadrant is shown next to it, and the great-circle distance in kilometers is shown below. The numbers refresh as you type.

Try 40.7128 N, -74.0060 W as the starting point and 35.6762 N, 139.6503 E as the destination. The page returns an initial bearing of about 332.99 degrees (NNW) and a distance of about 10,851.7 km, the rough course for a transpacific flight from New York to Tokyo.

Once you have the bearing, you can convert the same coordinates into UTM eastings and northings with Lat Long to UTM Calculator so the route lines up with a local-grid map.

These benefits show up whenever you need a single bearing and distance from a pair of coordinates, without firing up a desktop GIS package.

• • Skip the manual atan2 math: Computing the bearing by hand requires converting degrees to radians and unwrapping atan2. The page returns a single signed degree value.
• • Get bearing and distance in one step: Most online calculators only give one or the other. The page returns the initial bearing, the 8-point compass quadrant, and the great-circle distance in the same pass.
• • Validate the coordinate range automatically: Latitudes outside -90 to +90 and longitudes outside -180 to +180 are flagged before the formula runs, catching off-by-sign and off-by-decimal mistakes when coordinates are copied between map tools.
• • Work in the signed decimal format your map uses: Inputs and outputs use the signed-decimal convention used by Google Maps, GIS layers, and most GPS units, so the result can be pasted into the next tool in your workflow.
• • Handle the same-point and antipode edge cases cleanly: Identical coordinates return 0 degrees and 0 km instead of NaN, and antipodal coordinates return a half-circumference distance with a 0-degree bearing instead of dividing by zero.

For long routes the result is the initial heading at the start of the leg, not the heading you would hold at the midpoint or end. For short visible-landmark routes the drift is small enough that the initial bearing is good enough to set a compass and walk.

For a different kind of trajectory calculation that uses arcsine and arccosine on a measured ellipse, Angle Of Impact Calculator works out the angle at which a droplet struck a surface.

The formula is fixed, but the assumptions underneath the inputs and the shape of the route change how the result should be read.

• • The haversine formula assumes Earth is a perfect sphere. For geodetic-grade work, switch to Vincenty's formulae or a WGS84 ellipsoid library.
• • The page returns the initial heading. The actual heading along a great-circle route changes continuously, so pilots update the heading at every waypoint on intercontinental flights.

Magnetic declination is the most common reason a hand-held compass does not match the page's bearing; declination varies from about +20 degrees in eastern North America to about -10 degrees in Australia and drifts over decades, so add or subtract the local declination from the true-north bearing before you set a magnetic compass.

According to Omni Calculator, the initial azimuth from latitude and longitude is theta = atan2(sin(Delta_lambda) * cos(phi_2), cos(phi_1) * sin(phi_2) - sin(phi_1) * cos(phi_2) * cos(Delta_lambda)), and the great-circle distance uses the haversine formula with R = 6371 km.

If the bearing is the first step in choosing a solar orientation, Solar Panel Calculator takes the same roof azimuth and combines it with tilt and peak sun hours to size a working array.