Triangulation

Triangulation

A method of locating the position of an object by observing the direction and/or distance to the object from two or more observation points.

Key Formula

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

Where:

$a, b, c$ = Side lengths of the triangle opposite to angles A, B, C respectively
$A, B, C$ = Interior angles of the triangle at the vertices opposite sides a, b, c respectively

Worked Example

Problem: Two observers stand 500 m apart on flat ground (points A and B). They both spot a distant tower at point C. Observer A measures the angle from the baseline AB to the tower as 70°. Observer B measures the angle from the baseline BA to the tower as 65°. How far is the tower from observer A?

Step 1: Identify the known elements. The baseline AB = 500 m (side c, opposite angle C). Angle A = 70° and angle B = 65°.

c = 500 \text{ m}, \quad A = 70°, \quad B = 65°

Step 2: Find angle C using the triangle angle sum.

C = 180° - 70° - 65° = 45°

Step 3: Apply the Law of Sines to find side b (the distance from A to the tower C).

\frac{b}{\sin B} = \frac{c}{\sin C} \implies b = \frac{c \cdot \sin B}{\sin C}

Step 4: Substitute the known values and compute.

b = \frac{500 \times \sin 65°}{\sin 45°} = \frac{500 \times 0.9063}{0.7071} \approx 640.8 \text{ m}

Answer: The tower is approximately 641 m from observer A.

Frequently Asked Questions

How is triangulation different from trilateration?

Triangulation determines position by measuring angles from known points and then computing distances using triangle geometry. Trilateration determines position by measuring distances directly (for example, using signal travel times) from three or more known points. GPS, for instance, uses trilateration, not triangulation, because it measures distances via satellite signal timing.

Why do you need at least two observation points for triangulation?

A single angle measurement from one point only gives you a direction — a line extending outward. You need a second observation point (and its angle) to create two direction lines that intersect at a unique location. Without that second reference, the target could be anywhere along the first line.

Triangulation vs. Trilateration

Triangulation locates a point by measuring angles from a known baseline and computing the geometry of the resulting triangle. Trilateration locates a point by measuring distances from three or more known positions. Both solve for an unknown location, but they rely on different types of measurements. Surveying traditionally uses triangulation (angle-based), while GPS uses trilateration (distance-based).

Why It Matters

Triangulation is one of the oldest and most practical applications of trigonometry. Surveyors have used it for centuries to map land, measure the heights of mountains, and chart coastlines — all without needing to physically travel to every point. It also underpins navigation, astronomy, and modern technologies like cell tower positioning.

Common Mistakes

Mistake: Assuming GPS uses triangulation.

Correction: GPS uses trilateration, which measures distances from satellites via signal timing. Triangulation specifically involves angle measurement. The two terms are frequently confused in everyday language.

Mistake: Forgetting to compute the third angle before applying the Law of Sines.

Correction: In a typical triangulation problem, you know the baseline and two angles. You must first find the third angle (since angles in a triangle sum to 180°) before you can use the Law of Sines to solve for unknown sides.

Related Terms

Trigonometry — The branch of math underlying triangulation
Law of Sines — Key formula used to solve triangulation problems
Law of Cosines — Used when side-angle-side information is known
Triangle — The geometric shape formed in every triangulation
Angle — The primary measurement in triangulation
Sine — Trigonometric function central to angle-based calculations