Azimuth and distance between two points (inverse and direct problem)
Azimuth, bearing and distance between two points (inverse problem), or the other way round: the coordinates of a point from an azimuth and a distance (direct problem, side shot). In UTM (grid) or latitude/longitude (GRS80 ellipsoid: WGS 84, NAD83, ETRS89, GDA2020, SIRGAS 2000…).
In UTM, the azimuth is the grid azimuth (from grid North).
In lat/long, the azimuth is the true (geodetic) azimuth and the distance is on the ellipsoid.
Elevations (optional)
With the elevations you also get the height difference, the slope distance, the grade and the ground distance (local horizontal plane).
Result
Fill in the points to see the result.
Grid azimuth, true azimuth and bearing
Azimuth is the angle measured clockwise from North, from 0° to 360°. On the UTM grid it comes from atan2(ΔE, ΔN) and is the grid azimuth, counted from grid North. The true azimuth (geodetic) is counted from geographic North: the difference between the two is the meridian convergence, which we show as well.
The bearing (quadrant bearing) is the acute angle from the north–south line, 0° to 90°, with its quadrant: NE, SE, SW or NW. E.g. azimuth 200°30' = bearing S 20°30' W (shown here as 20°30' SW). The back azimuth is the azimuth from point 2 back to point 1 (azimuth ± 180° on the grid).
Grid vs. ground distance: the UTM grid distance is the ellipsoid distance multiplied by the scale factor k. The ground distance (local horizontal plane) is also corrected for elevation: dground ≈ dellipsoid × (R + h) ÷ R. Together they make the combined factor; on a job site the difference can exceed 1 meter per kilometer.
FAQ
What is the difference between the UTM distance and the distance measured with a tape or total station?
A total station measures on the ground (local horizontal plane). UTM is a projection: near the central meridian it shrinks distances (k = 0.9996) and near the zone edge it stretches them. On top of that, the higher the terrain, the longer the real distance compared with the ellipsoid. The difference is usually between about −0.4 and +1 m per kilometer. With the elevations filled in, the calculator shows all three distances (grid, ellipsoid and ground).
How do I compute the azimuth between two UTM coordinates by hand?
ΔE = E2 − E1, ΔN = N2 − N1. Azimuth = atan2(ΔE, ΔN), converted to degrees, plus 360° if negative. Distance = √(ΔE² + ΔN²). By quadrant: ΔE and ΔN positive → NE; ΔE positive and ΔN negative → SE; both negative → SW; ΔE negative and ΔN positive → NW.
What is the direct problem (side shot)?
It is the reverse: you have a known point, an azimuth and a distance, and you want the coordinates of the new point. In UTM: E2 = E1 + d·sin(Az) and N2 = N1 + d·cos(Az). In latitude/longitude we use Vincenty's formula on the GRS80 ellipsoid (WGS 84, NAD83, ETRS89, SIRGAS 2000…).
Which bearing and distance go into a legal description or plat?
It depends on the local rules. Some records use grid bearings and grid distances on a stated projection (UTM or a national/state grid), others true bearings and ground distances, or a local system. Always state the basis of bearings and whether distances are grid or ground; this tool gives the grid and true azimuths and, with the elevations, the ground distance.