How to use the dead reckoning calculator
- Enter the start latitude and longitude in decimal degrees (north and east positive, south and west negative).
- Enter the heading, the speed through the water with its unit, and the time underway in hours and minutes.
- Open Advanced options to add a current: the direction it flows toward and its speed, in the same speed unit.
- Read the estimated position, the distance run, the course and speed over the ground, and the check that works back to the start.
Formula
velocity over the ground = own velocity + current velocity (north and east parts added)
distance = speed over the ground x time 1 nautical mile = 1.852 km exactly
new latitude = asin(sin lat1 cos d + cos lat1 sin d cos course), new longitude from atan2, with d = distance / 6,371.0088 km
Dead reckoning estimates a position from a known start, a course and a distance. The sphere formulas follow a great circle from the start along the initial course. The page checks the result by finding the great circle distance back to the start.
Six knots at 065 degrees for four and a half hours
From 41.5 N, 70 W at a heading of 65 degrees and 6 knots for 4 hours 30 minutes, the distance run is 27 nautical miles (50.00 km, 31.07 miles). The estimated position is 41.68876 N, 69.45423 W, which is 41 degrees 41.326 minutes north and 69 degrees 27.254 minutes west. A great circle distance from there back to the start gives 27 nautical miles, which matches.
Adding a 2 knot current flowing toward 180 degrees (south) changes the course over the ground to 84.4 degrees at 5.46 knots and moves the estimate to 41.53887 N, 69.45549 W. The destination formula agrees with a separate calculation done by rotating vectors on the sphere.
Why dead reckoning drifts
Every error in heading, speed or time adds to the position error, and the error grows with each hour run. Currents and wind push a vessel sideways, so the course over the ground differs from the heading, as the optional current shows. Navigators correct the estimate whenever they get a fix from a landmark, a chart or a positioning system.
The Earth is modeled as a sphere of radius 6,371.0088 km. The real shape is slightly flattened, so over long distances the estimate differs from one on an ellipsoid by a small fraction of a percent.
Safety note
This page is for learning, planning and cross-checking. It must not be your only means of navigation, and it does not use magnetic variation, tides, leeway or charts. For distances and bearings between two known points, see the Great Circle Distance Calculator, and to switch between decimal degrees and degrees and minutes, the Coordinate Format Converter.
More navigation tools
For bearings, climbs and calendars, try the Elevation Profile and Grade Calculator, Solstice and Equinox Calculator and Bearing and Azimuth Converter.
Assumptions and limits
- Heading and current direction are true bearings, measured clockwise from true north.
- Speed and current are constant for the whole time underway.
- The Earth is a sphere with a mean radius of 6,371.0088 km.
- The current is added as a vector to the boat's own velocity; leeway and wind are not modeled.
- The example position and course are an illustration. Replace them with your own.
Frequently asked questions
What is the difference between heading and course over the ground?
Heading is where the bow points. Course over the ground is the direction the boat actually travels once the current is added. They match only when there is no current or the current flows along the heading.
Does it work near the poles or across the date line?
Yes. Longitude is wrapped to between -180 and 180 degrees, and the formulas also work over a pole, though a heading is not meaningful at exactly 90 degrees latitude.
Why is longitude shown in degrees and decimal minutes?
Charts and many navigation instruments use that form. The decimal degree values are shown beside it so you can enter them in a map or a coordinate converter.
How accurate is the result?
It is only as good as the inputs. The sphere model adds an error of a fraction of a percent of the distance, and the main error in practice comes from the heading, speed and current you enter.