About Bearing & Distance (COGO) Calculator
The bearing and distance calculator solves the two classic coordinate geometry (COGO) problems of plane surveying. The inverse problem takes two points and returns the horizontal distance between them and the azimuth (whole-circle bearing) of the line from the first point to the second. The forward problem takes a starting point, a bearing, and a distance and returns the coordinates of the end point.
All directions use the surveyor’s azimuth convention: a whole-circle bearing measured clockwise from North, from 0 up to 360 degrees. Because the azimuth is referenced to the Northing axis and turns toward Easting, it is computed as atan2(ΔEasting, ΔNorthing). The tool reports the azimuth in both decimal degrees and degrees-minutes-seconds and also gives the quadrant (reduced) bearing such as N 36°52′ E.
How It Works
- Choose the inverse mode (two points) or the forward mode (point, bearing, distance).
- For the inverse problem the calculator forms ΔE = E2 − E1 and ΔN = N2 − N1, then distance = sqrt(ΔE² + ΔN²) and azimuth = atan2(ΔE, ΔN) normalised to 0–360°.
- For the forward problem it computes E2 = E1 + L·sin(azimuth) and N2 = N1 + L·cos(azimuth).
- The azimuth is converted to degrees-minutes-seconds and classified into a quadrant (NE, SE, SW, NW) with a reduced angle between 0 and 90 degrees for the whole-circle-to-quadrant bearing.
Worked Example
Inverse problem: from station A at (E = 1000, N = 2000) to station B at (E = 1300, N = 2400), ΔE = 300 and ΔN = 400, so the distance is sqrt(300² + 400²) = 500.000 m. The azimuth is atan2(300, 400) = 36.8699°, which is 36°52′11.6″, in the NE quadrant, giving a quadrant bearing of N 36°52′ E. Running the forward problem from A with that bearing and 500 m returns B exactly.
Formulas
- Inverse — distance
distance = sqrt( (E2 - E1)^2 + (N2 - N1)^2 )- Inverse — azimuth (clockwise from North)
azimuth = atan2( (E2 - E1), (N2 - N1) ), normalised to 0..360 deg- Forward — new coordinates
E2 = E1 + L * sin(azimuth); N2 = N1 + L * cos(azimuth)- Quadrant (reduced) bearing
NE: 0–90 -> N θ E | SE: 90–180 -> S (180−az) E | SW: 180–270 -> S (az−180) W | NW: 270–360 -> N (360−az) W
Standards & References
- Coordinate geometry (COGO)
- Plane surveying azimuth convention (clockwise from North)
- Ghilani & Wolf, Elementary Surveying
Frequently Asked Questions
How is the azimuth measured and why is it atan2(ΔE, ΔN)?
The azimuth is a whole-circle bearing measured clockwise from North, 0 to 360 degrees. Because North is the Northing axis and the angle turns toward Easting, the direction is atan2(ΔEasting, ΔNorthing), which is the transpose of the usual mathematical atan2(y, x) used for angles measured counter-clockwise from East.
What is the difference between a whole-circle bearing and a quadrant bearing?
A whole-circle bearing (azimuth) is a single angle from 0 to 360 degrees measured clockwise from North. A quadrant or reduced bearing expresses the same direction as an angle between 0 and 90 degrees relative to North or South, with an E or W suffix, for example N 36°52′ E for an azimuth of 36.87°.
What is the difference between the inverse and forward problems?
The inverse problem starts from two known coordinates and computes the distance and bearing between them, which is how you derive directions from a survey. The forward problem starts from one point with a known bearing and distance and computes the coordinates of the new point, which is how you set out or lay down stations.
Which coordinate is Easting and which is Northing?
Throughout the tool the first value is the Easting (the x or horizontal coordinate) and the second is the Northing (the y or vertical coordinate). Distances are plane horizontal distances in the same length unit as the coordinates, typically metres.