Take a relative bearing on a landmark, hold course and speed, take a second bearing, and enter both with the time between them. The page returns the run, your distance off at the second bearing, and how close you will pass.
Use this when you have one charted landmark, a steady course, and a known speed — the classic running fix with relative bearings measured from the bow.
Relative bearings are angles from the bow to the object, on either side, between 0° and 180°. The second bearing must be larger than the first (the object draws aft as you pass). Speed should be speed over ground if you know it; through-the-water speed works only when there is no current.
Special cases are built in: doubling the angle on the bow (second bearing twice the first) makes the distance off equal to the run, and bow-and-beam (45° then 90°) gives the abeam distance directly.
The two bearing lines and your track form a triangle. The side along your track is the run (speed × time). The angle at the first position is the first relative bearing A; the angle at the second position is 180° − B; so the angle at the landmark is B − A. The law of sines gives the distance from the second position to the landmark: run × sin A ÷ sin(B − A).
The passing distance — how far off the landmark is when it comes abeam — is the perpendicular from the landmark to your track: the second distance times sin B. When B is exactly twice A, the triangle is isosceles and the distance off equals the run, which is why “doubling the angle on the bow” is a favorite trick.
run = speed × minutes / 60
d₂ = run × sin A / sin(B − A)
d₁ = run × sin B / sin(B − A)
passing distance = d₂ × sin B
Each row below starts from the default inputs and changes only the columns shown. Recompute any row by hand with the formula above.
| A | B | Knots | Min | Run nm | Off at B | Abeam |
|---|---|---|---|---|---|---|
| 45 | 90 | 6 | 20 | 2.00 | 2.00 | 2.00 |
| 30 | 60 | 6 | 20 | 2.00 | 2.00 | 1.73 |
| 22.5 | 45 | 6 | 20 | 2.00 | 2.00 | 1.41 |
| 26.5 | 45 | 6 | 20 | 2.00 | 2.81 | 1.99 |
| 35 | 70 | 5 | 30 | 2.50 | 2.50 | 2.35 |
| 40 | 110 | 7 | 15 | 1.75 | 1.20 | 1.12 |
With the calculator defaults (First relative bearing (° from bow) = 45; Second relative bearing (° from bow) = 90; Speed (knots) = 6; Time between bearings (min) = 20), the arithmetic runs like this:
run = speed × minutes / 60 = 6 × 20 / 60 = 2.000 nm
distance at 2nd bearing = run × sin A / sin(B − A) = 2.000 × sin 45° / sin 45° = 2.000 nm
passing distance = d₂ × sin B = 2.000 × sin 90° = 2.000 nm
Result: 2.00 nm off at 2nd bearing. Run 2.00 nm. Distance off at the first bearing 2.83 nm. Passing (abeam) distance 2.00 nm. Doubled angle: distance off at the second bearing equals the run.
Now change First relative bearing (° from bow) to 30 and leave everything else alone:
run = speed × minutes / 60 = 6 × 20 / 60 = 2.000 nm
distance at 2nd bearing = run × sin A / sin(B − A) = 2.000 × sin 30° / sin 60° = 1.155 nm
passing distance = d₂ × sin B = 1.155 × sin 90° = 1.155 nm
Result: 1.15 nm off at 2nd bearing. Run 2.00 nm. Distance off at the first bearing 2.31 nm. Passing (abeam) distance 1.15 nm.
With bearings of 30° and 60° the angle is doubled again, so the distance off at the second bearing still equals the 2-nm run — but the passing distance shrinks to about 1.73 nm because the landmark is still ahead of the beam.
Two bearing pairs are worth remembering: 22.5° and 45° gives a passing distance of about 7/10 of the run, and 26.5° and 45° gives a passing distance about equal to the run. Both let you predict how close you will pass a headland long before it comes abeam.
Current and leeway are the weak point. If the boat is being set across its course, the run along the water is not the run over the ground, and the triangle is wrong. Use speed over ground and the course actually made good when you have them.
Take bearings with the same instrument each time — a hand-bearing compass or a pelorus — and note the exact time. A one-minute timing error at 6 knots is 0.1 nm of run.
Units: Degrees for bearings, knots and minutes for the run, nautical miles for distances.
Where the model stops: Constant course and speed over ground between the bearings, flat-chart geometry. Current, leeway, and bearing errors are not modeled.
Not a substitute: A running fix is a lower-confidence position than a simultaneous cross-bearing fix. Cross-check with other aids and keep a safe margin off hazards.
| First / second bearing | Distance off at 2nd bearing | Passing distance |
|---|---|---|
| 22.5° / 45° | 1.000 × run | 0.707 × run |
| 26.5° / 45° | 1.406 × run | 0.994 × run |
| 30° / 60° | 1.000 × run | 0.866 × run |
| 45° / 90° | 1.000 × run | 1.000 × run |
| 30° / 90° | 0.577 × run | 0.577 × run |
| 60° / 90° | 1.732 × run | 1.732 × run |
Doubling the angle (22.5°/45°, 30°/60°, 45°/90°) always makes the distance at the second bearing equal to the run. The 26.5°/45° pair gives a passing distance almost exactly equal to the run, which tells you in advance how far off you will pass.
The angle from the bow to the object, measured either side. An object dead ahead is 0°, abeam is 90°, dead astern is 180°.
When the second bearing is twice the first, the angle at the landmark equals the first bearing, so the triangle has two equal angles and two equal sides: the run and the distance off.
Yes, if you convert them to angles relative to your course: subtract the course from each bearing (and take the absolute value, keeping it between 0° and 180°).
The geometry is the same. Enter the angles from the bow without a sign.