Bearing and distance from coordinates: survey inverse calculator

Updated: September 30, 2026 · ISKRIT

The inverse computation gives the distance and bearing between two points from their coordinates: distance = √(ΔN² + ΔE²) and the bearing is the angle whose tangent is ΔE / ΔN, placed in the right quadrant by the signs of the differences. The forward computation goes the other way and gives the coordinates of a new point from a bearing and a distance.

Distance and bearing between two points

Coordinates from bearing and distance

How it works

Differences. ΔN = NB - NA and ΔE = EB - EA. Northing runs to grid north and easting to grid east, and bearings are measured clockwise from grid north.

Quadrant. The acute angle arctan(|ΔE| / |ΔN|) is the quadrant bearing. The signs of the differences name it and convert it to a whole-circle bearing.

Check. The back bearing differs from the forward bearing by exactly 180°. The distance can be checked twice as ΔN / cos(bearing) and ΔE / sin(bearing).

Forward computation. ΔN = distance · cos(bearing), ΔE = distance · sin(bearing). The signs of the sine and cosine place the new point in the right quadrant.

Worked example

A (N 6210300.00, E 512040.00), B (N 6210351.00, E 512035.20). ΔN = +51.000 and ΔE = -4.800, so the line runs north-west. Distance = √(51.000² + 4.800²) = 51.225. θ = arctan(4.800 / 51.000) = 5°22′36″, the quadrant bearing is N 5°22′36″ W and the whole-circle bearing is 360° - θ = 354°37′24″. The forward computation from A with this bearing and distance returns B to the millimetre.

Questions

Grid bearing or true bearing? Coordinates on a projected grid give grid bearings. True bearings differ by the grid convergence at the point.

Which units? Any, as long as both points use the same: the distance comes out in the units of the coordinates.

Why is my bearing off by 90° or 180°? Northing and easting are usually swapped. The calculator expects northing first for each point.

Deviations of set-out and as-built points from design coordinates, with a drawing, are computed by the as-built survey calculator.