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Instant Calculation
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Triangulation locates an unknown point C using a known baseline between two fixed reference points A and B, plus the angles that lines AC and BC make with that baseline — a classic surveying and navigation technique that requires no direct distance measurement to the unknown point at all, only two angle readings and a known baseline distance. It's the working principle behind cell-tower and GPS-adjacent location estimation, land surveying (locating an inaccessible landmark, like a mountain peak, from two known survey markers), and celestial navigation and astronomy (measuring stellar parallax uses this exact same idea to find a star's distance from two observation points).
How it works
Enter baseline points A and B, plus the bearing angles measured at A and at B (both measured counterclockwise from the baseline direction, with the unknown point C assumed to lie on the same side of the baseline for both readings). The calculator first finds the baseline's length and direction from A and B's coordinates. Since the two given angles plus the triangle's third angle must sum to 180°, the angle at C follows immediately (180° − angleA − angleB). The Law of Sines, AC/sin(B) = AB/sin(C), then gives the distance AC. Finally, C's coordinates are found by starting at A and projecting a distance of AC along the direction angleA measured from the baseline.
- Enter baseline point A — x.
- Enter baseline point A — y.
- Enter baseline point B — x.
- Enter baseline point B — y.
- Enter angle at A, from baseline to C (degrees).
- Enter angle at B, from baseline to C (degrees).
- Click Calculate to see your results.
Examples
A(0,0), B(10,0), both angles 45°
This is a 45-45-90 triangle. Angle C = 180−45−45 = 90°. AC = 10×sin(45°)/sin(90°) ≈ 7.071. Projecting from A at 45°: C = (7.071×cos45°, 7.071×sin45°) = (5, 5) — directly above the baseline's midpoint, as expected by symmetry.
A(0,0), B(20,0), angle A=30°, angle B=60°
Angle C = 180−30−60 = 90°. AC = 20×sin(60°)/sin(90°) ≈ 17.32. Projecting from A at 30°: C = (17.32×cos30°, 17.32×sin30°) ≈ (15.0, 8.66). Checking via B: BC = 20×sin(30°)/sin(90°) = 10, and the distance from B(20,0) to (15.0, 8.66) is √(5²+8.66²) ≈ 10.0 — confirming the result.
A(0,0), B(8,0), angle A=50°, angle B=70°
Angle C = 180−50−70 = 60°. AC = 8×sin(70°)/sin(60°) ≈ 8.68. Projecting from A at 50°: C ≈ (8.68×cos50°, 8.68×sin50°) ≈ (5.58, 6.65).
Who should use it
- Surveying an inaccessible landmark from two known reference points.
- Estimating a device or signal source's location from two known bearing measurements.
- Astronomy problems involving stellar parallax and distance estimation.
Industry applications
- Land surveying and geodesy
- Navigation and radio direction-finding
- Astronomy
Advantages
- Locates a point using only two angle measurements and a known baseline — no direct distance measurement to the target needed.
- Directly applicable to real surveying, navigation, and astronomy problems.
Limitations
- Accuracy degrades for very "thin" triangles where the angles are close to 0° or 180°.
Common mistakes to avoid
- Measuring the angles from the wrong reference direction — both must be measured counterclockwise from the baseline, on the same side as the target point.
- Forgetting to check that angleA + angleB is strictly less than 180° before proceeding.
- Confusing triangulation (uses angles) with trilateration (uses distances) — they solve a similar problem with fundamentally different input data.
Best practices
- Choose a baseline and viewing angles that produce a well-proportioned triangle (avoiding angles very close to 0° or 180°) to minimize measurement error amplification.
- Double check both angles are measured from the same consistent reference direction and side of the baseline.
- Cross-verify your computed point by checking the distance from both A and B against the Law of Sines results.
Tips
- Need to verify three raw side lengths form a valid triangle before further analysis? Use the Triangle Inequality Theorem Calculator.