Math
Triangulation Calculator
Give the coordinates of two observation points and the angle at which the target is seen from each, either as compass bearings (clockwise from north) or as direction angles (counterclockwise from the +x axis). mathcheck converts the angles, finds the interior angles of the observation triangle, applies the law of sines to get the distance from each observer, and reports the target's coordinates with a warning when the sight lines cross at a shallow angle or behind an observer.
In degrees, using the convention chosen above.
In degrees, using the convention chosen above.
Try an example
Result
Target position
—
- Distance from A
- —
- Distance from B
- —
- Baseline AB
- —
- Angle at A
- —
- Angle at B
- —
- Angle at the target
- —
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Study path
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How triangulation works
If you know where two observers stand and the direction each of them sees a target, the target must lie where those two sight lines cross. The observers and the target form a triangle whose base (the baseline) is known and whose base angles come from the sightings, so the law of sines gives the remaining sides and hence the target's position. Surveyors, navigators, and hikers with a map and compass all use this idea.
Observers at (0, 0) and (10, 0), bearings 45° and 315°
Direction angles are 45° from A and 135° from B. Both base angles are 45°, so γ = 90° and AP = 10 · sin 45° / sin 90° = 7.07. The target is at (0 + 7.07 cos 45°, 7.07 sin 45°) = (5, 5).
| Angle at the target | Quality of the fix |
|---|---|
| near 90° | best: bearing errors move the fix the least |
| 30° to 150° | acceptable |
| under 10° or over 170° | poor: the lines are nearly parallel and tiny errors shift the fix far |
| 0° or 180° | no fix: the lines never cross |
How to use this calculator
- Enter the coordinates of both observation points (any consistent units, such as meters or map grid units).
- Choose whether your angles are compass bearings or mathematical direction angles.
- Enter the angle to the target from each observer.
- Read the target's coordinates, the distance from each observer, and the triangle's three angles.
- Open Show the work to see the bearing conversion, the interior angles, and the law-of-sines computation.
How to read the answer
The target position is exact to floating-point precision, but its practical accuracy depends on the sighting angles. The angle at the target measures the quality of the intersection: close to 90° is ideal. A warning that the lines cross behind an observer usually means a bearing was recorded 180° off (a back bearing). Distances come out in the same units as the coordinates.
Common mistakes and edge cases
- Mixing bearings and direction angles; a bearing of 45° is a direction angle of 45° only by coincidence, and 90° bearing is 0° direction.
- Reading a back bearing (the direction from the target to the observer) instead of the forward bearing, which is off by 180°.
- Using two observation points that line up with the target, which makes the sight lines nearly parallel.
- Forgetting that the interior angles at A and B are measured from the baseline, not from north.
- Assuming the fix is exact. Each degree of bearing error moves the fix about 1.7% of the distance sideways.
Worked examples
Symmetric fix
A(0, 0), B(10, 0), bearings 45° and 315°
Target position
(5, 5)
Direction angles
A(0, 0), B(10, 0), directions 60° and 120°
Target position
(5, 8.6603)
Off-axis baseline
A(1, 2), B(7, 5), directions 90° and 180°
Target position
(1, 5)
Target south of the baseline
A(0, 0), B(10, 0), bearings 135° and 225°
Target position
(5, -5)
Shallow intersection
Bearings 5° and 355°: the lines cross far away
Target position
(5, 57.1503)
Back bearing mistake
Bearings 225° and 135° cross behind both observers
Target position
(5, 5)
Parallel sight lines
Target position
—
Same observation point
Target position
—
Frequently asked questions
What is triangulation?+
Finding an unknown position from two known positions and the directions to the target from each. The target sits where the two sight lines intersect, and the law of sines gives the distances.
What is the difference between a bearing and a direction angle?+
A compass bearing is measured clockwise from north; a direction angle is measured counterclockwise from the positive x-axis (east). They are related by direction = 90° - bearing.
Why does the calculator warn about a thin intersection?+
When the sight lines cross at a small angle, a tiny bearing error slides the crossing point a long way along the lines. Fixes with an angle near 90° at the target are the most reliable.
What does 'the lines cross behind an observer' mean?+
The mathematical intersection lies in the opposite direction from where that observer was looking. In practice it means one bearing was taken backwards or misread by 180°.
Can I triangulate with three observers?+
Yes, pairwise. Three sight lines rarely meet at a single point; they form a small triangle (the cocked hat), and its size indicates the measurement error. This page handles one pair at a time.
Does triangulation need distances?+
No, only angles. That is what distinguishes it from trilateration (used by GPS), which uses distances from known points instead of directions.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026