Triangulation Calculator
Enter two known station coordinates and either the bearing from each station or the distance from each station. The calculator solves for the unknown point using the intersection (bearing-bearing) method or the trilateration (distance-distance) method, shows step-by-step working, and draws a live shape diagram.
Formula
Worked example
Intersection example: A=(0,0), B=(800,0), bearing from A=45 deg (NE), bearing from B=315 deg (NW). Direction vectors: A->(0.707,0.707), B->(-0.707,0.707). Denominator=0.707*0.707-0.707*(-0.707)=1.0. t=((800)(0.707)-(0)(-0.707))/1.0=565.7. P=(0+565.7*0.707, 0+565.7*0.707)=(400,400). The point lies exactly midway above the baseline, as expected by symmetry.
What is triangulation?
Triangulation is a technique for locating an unknown point by measuring angles from two or more known reference positions. The word comes from the Latin for triangle: you form a triangle with the two known stations and the unknown point, then use the measured bearings or distances to pin down the missing vertex. Surveyors, navigators, and engineers have used this method for centuries - it underpins geodetic mapping, search-and-rescue positioning, and modern GPS. The method that uses angles is called intersection (or angular triangulation); the version that uses distances is trilateration, though the word triangulation is often used loosely for both.
Intersection method - bearing from two known stations
In the intersection method you stand at two known points A and B and measure the bearing (azimuth) from each toward the target P. Each bearing defines a ray from A and B respectively. The target lies where those two rays cross. Mathematically, each ray is a parametric line: for station A with coordinates (Ax, Ay) and bearing theta_A, the ray is (Ax + t*cos(theta_A), Ay + t*sin(theta_A)). Setting the two ray equations equal and solving for the parameter t gives the intersection coordinates. The solution fails (the rays are parallel) only if both bearings aim in exactly the same direction, which means A and P and B are collinear. For best accuracy the intersection angle at P (the interior angle of the triangle at the unknown vertex) should be between 30 and 150 degrees.
Trilateration method - distance from two known stations
When you know your distance from two stations instead of your bearing from them, you use trilateration. Each distance defines a circle centred on the known station. The unknown point lies where the two circles intersect, which gives two candidate positions (one on each side of the baseline). This calculator picks the candidate with the higher Y (Northing) coordinate, which is the conventional default; real-world practice uses additional information (a third measurement, or knowledge of which side of the baseline is correct) to confirm the right solution. The analytic formula first solves the problem in a rotated frame where A is the origin and B lies on the positive x-axis, then rotates back to world coordinates. The formula for the local x-position is x = (AB^2 + rA^2 - rB^2) / (2 * AB), and the local y is sqrt(rA^2 - x^2).
Derived outputs - distances, bearings and area
Beyond the coordinates of the solved point, this calculator reports the distance from each station to P, the bearing from each station to P, the interior angle at P, and the area of the triangle formed by A, B and P. The bearings are computed as four-quadrant inverse tangents converted to azimuths (0-360 degrees clockwise from North). The interior angle at P comes from the law of cosines applied to the three side lengths. The triangle area uses the shoelace (cross-product) formula: half the absolute value of (AB_x * AP_y - AP_x * AB_y), where AB and AP are vectors from A to B and A to P respectively.
Intersection angle quality guide
| Angle at P | Geometry quality | Practical accuracy |
|---|---|---|
| Less than 15 deg | Poor | Small errors produce large position shifts |
| 15 - 29 deg | Marginal | Use only when better geometry is unavailable |
| 30 - 59 deg | Acceptable | Suitable for most survey tasks |
| 60 - 120 deg | Excellent | Optimal zone; error propagation is minimised |
| 121 - 150 deg | Acceptable | Suitable for most survey tasks |
| Greater than 150 deg | Marginal | Similar to very acute angles - errors amplify |
The angle formed at the unknown point P affects how sensitive the position is to small bearing or distance errors.
Frequently asked questions
What is the difference between triangulation and trilateration?
Triangulation uses angle (bearing) measurements from known stations to locate an unknown point. Trilateration uses distance measurements instead. Both form a triangle and solve for the unknown vertex, but the math is different. GPS uses trilateration: it measures travel-time distances to satellites, not angles.
Why does the calculator need two stations?
One station gives only a line (a single bearing) or a circle (a single distance), leaving the target's position uncertain. Two stations produce two lines (bearing-bearing) or two circles (distance-distance) whose intersection pins down the unique point. A third station is often added in practice to confirm accuracy and detect errors.
What does the intersection angle at P mean, and why does it matter?
The intersection angle is the interior angle of the triangle at the unknown point P. When this angle is close to 0 or 180 degrees the two rays (or circles) meet almost tangentially, so a tiny error in a bearing or distance shifts the computed position by a large amount. Angles between 30 and 150 degrees keep error propagation small; the ideal is near 90 degrees.
Trilateration returns two solutions - how do I know which is correct?
Two circles always intersect at two points on opposite sides of the baseline. This calculator picks the solution with the higher Y coordinate as a default. In real fieldwork you eliminate the wrong solution using a third measurement, terrain knowledge, or a rough compass bearing to confirm which side of the baseline the target lies on.
What coordinate system should I use?
Any consistent planar coordinate system works: local grid coordinates, a national grid like UTM or the British National Grid, or simply a site-specific grid in metres or feet. The calculator does not assume a geographic (lat/lon) projection, so you must convert lat/lon to a flat grid before entering coordinates. For long distances, the curvature of the Earth makes a flat-grid assumption less accurate.
Can I use this for GPS or geographic coordinates?
Only for small areas. The formulas assume a flat plane, which is valid for a few kilometres. For larger areas or precise geodetic work, convert your geographic coordinates to a projected system (such as UTM) first, perform the triangulation in metres, then convert back. Dedicated geodetic software handles the ellipsoid corrections automatically.