Find an unknown position using triangulation (angles) or trilateration (distances) from 2-3 known reference points. Applications in GPS, surveying, and navigation.
Triangulation and trilateration are the two fundamental methods for determining an unknown position from known reference points. Triangulation uses measured angles from two known points to compute the target's location using the law of sines. Trilateration uses measured distances from two or three known points and finds the target by intersecting circles.
GPS navigation uses trilateration: each satellite provides a distance (from signal timing), and three or more intersecting spheres pinpoint your location. Land surveying traditionally uses triangulation: a surveyor measures angles to a target from two benchmarks with known coordinates and a known baseline distance. Both methods produce exact solutions in ideal conditions, with the third reference point (in trilateration) resolving the ambiguity between two possible positions.
This calculator supports three modes: two-point trilateration (which yields two candidate positions), three-point trilateration (which resolves the ambiguity), and two-point triangulation (using angles). Enter reference point coordinates, distances or angles, and get the target position, reference point summary, and distance visualization. Presets for GPS, surveying, and cell tower scenarios let you explore immediately.
Solving trilateration by hand requires intersecting circles — a process involving quadratic equations and careful algebra. Triangulation requires trigonometric computation with the law of sines. Both methods involve multiple steps where arithmetic errors are common, especially with non-integer coordinates.
This calculator handles the entire computation instantly, supports three different methods, shows both candidate solutions for ambiguous cases, and provides visual feedback on the solution quality. It is ideal for students learning coordinate geometry, surveyors checking field calculations, and anyone exploring positioning systems.
Trilateration: intersect circles (x−x₁)²+(y−y₁)²=r₁² and (x−x₂)²+(y−y₂)²=r₂². Solve the linear equation from subtracting to find one coordinate, substitute for the other. Triangulation: use sine rule — d₁ = b·sin(β)/sin(γ), where b = baseline, β = angle at P2, γ = 180°−α−β.
Result: Target ≈ (2.45, 4.35), Residual error ≈ 0.04
Three circles centered at (0,0) r=5, (10,0) r=6, and (5,8) r=5 are intersected. The first two circles give two candidates; the third circle selects the one with smallest distance error.
The Global Positioning System (GPS) is the world's most widely used trilateration system. Each GPS satellite broadcasts its position and a precise timestamp. Your receiver measures the time delay of each signal, converting it to a distance (speed of light × time). With distances from three satellites, your 2D position is determined; with four, your 3D position plus clock error are solved simultaneously.
Modern GPS achieves meter-level accuracy in standard mode and centimeter-level with differential corrections (RTK-GPS). The mathematical core is the same circle/sphere intersection implemented in this calculator, extended to three dimensions and augmented with sophisticated error correction.
Before GPS, land surveying relied on triangulation networks: a web of points with precisely measured angles and a few precisely measured baselines. The Great Trigonometrical Survey of India (1802–1871) used this method to map the entire subcontinent, including measuring the height of Mount Everest. The mathematical principles are identical to this calculator's angle-based method, scaled to spherical geometry.
Beyond GPS, triangulation and trilateration appear in: Wi-Fi positioning (using signal strength as a proxy for distance), emergency call location (cell tower trilateration), robot localization (using laser rangefinders or beacons), earthquake seismology (locating the epicenter from seismic wave arrival times), and acoustic source localization (finding a sound source from microphone arrays).
Triangulation determines position from angle measurements; trilateration determines position from distance measurements. Both use known reference points but different input data.
Two satellites give two possible positions (the intersection of two spheres is a circle in 3D, or two points in 2D). A third satellite resolves the ambiguity. In practice, 4 satellites are needed because the clock offset introduces a fourth unknown.
If the circles don't intersect, the measured distances are inconsistent — the target cannot be at the specified distances from all reference points simultaneously. This indicates measurement error.
Yes. With more than 3 points, the system is over-determined, and a least-squares approach can be used to find the best-fit position, which also reveals measurement quality.
Accuracy depends on measurement precision and the geometry of reference points. A "well-conditioned" configuration (reference points spread around the target) gives better results than a "poorly conditioned" one (all on one side).
Geometric Dilution of Precision (GDOP) measures how satellite geometry affects position accuracy. When satellites are clustered together, GDOP is high (poor accuracy). When spread across the sky, GDOP is low (good accuracy).