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Triangulation Calculator

Determine a point's position from two known points and the angles to the unknown point using triangulation.

Reviewed by Christopher FloiedPublished Updated

This free online triangulation calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

x coordinate of station A

y coordinate of station A

x coordinate of station B

y coordinate of station B

Angle from baseline AB to the target, measured at station A

Angle from baseline BA to the target, measured at station B

Results

Baseline Distance

10

Distance A to Target

8.9658

Target x Coordinate

6.3397

Target y Coordinate

6.3397

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Triangulation Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Triangulation Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Triangulation Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

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About Triangulation Calculator

Triangulation is a method for determining the position of an unknown point by measuring angles from two known reference points (stations). By knowing the distance between the two stations (the baseline) and the angles from each station to the target, the law of sines determines the distances to the target, and trigonometry gives the coordinates. This technique has been used for centuries in surveying, navigation, and astronomy. Triangulation networks mapped entire countries before GPS existed — the Great Trigonometric Survey of India used triangulation to measure the Himalayas. Modern applications include cell tower positioning, where a phone's location is estimated from signal angles at multiple towers, and in robotics, where triangulation helps determine object positions using stereo cameras. In emergency services, triangulation of radio signals helps locate distress beacons. The mathematical foundation is the law of sines, which relates the sides and angles of a triangle. This calculator takes two station coordinates and two measured angles, then computes the target position using the sine rule and trigonometric direction calculations.

The Math Behind It

Triangulation constructs a triangle from a known base and two angles. Given stations A and B with a baseline of known length d_AB and measured angles α (at A) and β (at B) to an unknown target P, the angle at P is γ = π - α - β. By the law of sines, d_AP/sin(β) = d_AB/sin(γ), so d_AP = d_AB × sin(β)/sin(γ). Similarly, d_BP = d_AB × sin(α)/sin(γ). The position of P is then P = A + d_AP × (cos(θ_AB + α), sin(θ_AB + α)), where θ_AB = atan2(y_B - y_A, x_B - x_A) is the direction angle of the baseline. For triangulation to work, both angles must be positive and their sum must be less than 180°. When α + β is close to 180°, the target is far away and the result is very sensitive to small angle errors. The optimal configuration has the target roughly perpendicular to the baseline, making α + β close to 90°. The dilution of precision (DOP) concept quantifies this sensitivity. In three dimensions, triangulation requires measurements from at least three non-collinear stations or a combination of angle and distance measurements. Resection is the inverse problem: determining one's own position by measuring angles to three known points.

Formula Reference

Law of Sines

d_AP / sin(B) = baseline / sin(C)

Variables: d_AP = distance from A to target; B = angle at B; C = angle at target = π - A - B

Target Position

P = A + d_AP × (cos(θ_AB + α), sin(θ_AB + α))

Variables: θ_AB = baseline angle; α = angle at A; d_AP = distance from A to P

Worked Examples

Example 1: Triangulation from two stations

Station A at (0, 0), Station B at (10, 0). Angle at A = 45°, angle at B = 60°. Find the target.

Step 1:Baseline distance = 10
Step 2:Angle at target γ = 180° - 45° - 60° = 75°
Step 3:d_AP = 10 × sin(60°)/sin(75°) = 10 × 0.8660/0.9659 ≈ 8.9658
Step 4:Baseline angle θ_AB = atan2(0, 10) = 0
Step 5:Target x = 0 + 8.9658 × cos(0 + 45°) ≈ 6.3397
Step 6:Target y = 0 + 8.9658 × sin(45°) ≈ 6.3397

The target is at approximately (6.34, 6.34).

Example 2: Target directly above the baseline

A at (0, 0), B at (8, 0). Angle A = 90°, angle B = 45°.

Step 1:γ = 180° - 90° - 45° = 45°
Step 2:d_AP = 8 × sin(45°)/sin(45°) = 8
Step 3:Target x = 0 + 8 × cos(0 + 90°) = 0
Step 4:Target y = 0 + 8 × sin(90°) = 8

The target is at (0, 8), directly above station A.

Common Mistakes & Tips

  • !Not converting angles from degrees to radians before using trigonometric functions.
  • !Having angles sum to 180° or more, which means the triangle is degenerate or the angles are inconsistent.
  • !Forgetting to account for the baseline direction — angles must be measured relative to the baseline, not relative to north or the x-axis.
  • !Confusing triangulation (using angles) with trilateration (using distances, as in GPS).

Related Concepts

Intersection of Two Lines

Triangulation is geometrically equivalent to finding where two lines of sight intersect.

Polar Coordinates

The distance and angle from a station to the target is essentially a polar coordinate.

Three-Dimensional Distance

3D triangulation extends this concept to spatial positioning.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

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Frequently Asked Questions

What is the difference between triangulation and trilateration?

Triangulation uses measured angles from known positions to determine an unknown point. Trilateration uses measured distances. GPS uses trilateration (measuring distances to satellites), not triangulation, despite common misconception.

What happens if the angles do not sum correctly?

If angle A + angle B ≥ 180°, the triangle is impossible. If the sum is close to 180°, the target is very far away and the calculation is extremely sensitive to measurement errors.

How accurate is triangulation?

Accuracy depends on the precision of angle measurements and the geometry. Targets near the perpendicular bisector of the baseline give the best accuracy. Targets far away or nearly in line with the baseline have poor accuracy.

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