How this instrument works
Triangulation recovers the distance to a point that can't be measured directly — across a river, out to a ship, up to a distant landmark — using only a baseline between two observation points and the angles each point sights toward the target. With a baseline length and the angles at each end (A and B), the third angle follows from the Triangle Sum Theorem, C = 180° − A − B, and the Law of Sines then gives the distance from point A to the target: distance = baseline·sinB ⁄ sinC.
This is the same technique that has determined mountain heights, coastal distances, and even the scale of the solar system historically, long before laser rangefinders or GPS existed — a genuinely measurable baseline and two sighted angles are all that's ever needed, with trigonometry filling in the unmeasurable distance.
The baseline itself can be as short as convenient (the width of a survey plot) or as long as needed (the distance between two observatories, for astronomical triangulation) — what matters is that both endpoints can sight the same distant target, and that the baseline's own length is directly measurable, unlike the target distance itself.
- Enter the measured distance between the two observation points into the Baseline length field.
- Enter the angle sighted toward the target from point A into the Angle at point A field.
- Enter the angle sighted toward the target from point B into the Angle at point B field.
- Read Distance from A to the target: the sheet applies the Law of Sines to the resulting triangle.
Worked example — a 100-unit baseline, 50° and 60°
A surveyor measures a 100-unit baseline between two points, sighting the target at 50° from point A and 60° from point B. The third angle is 180−50−60=70°, and the distance from A to the target is 100·sin(60°) ⁄ sin(70°) ≈ 92.16 — recovered entirely from a measurable baseline and two sighted angles, with no direct measurement of the actual target distance ever needed.
A shorter 50-unit baseline with equal 45° angles at both ends gives a distance of about 35.36 — a symmetric setup, with the target sitting directly above the baseline's midpoint. A 200-unit baseline with an asymmetric 30° and 100° pairing gives a distance of about 257.12, a much wider spread reflecting the sharper sighting angles involved.
Questions
What is triangulation?
A technique for finding the distance to a point that can't be measured directly, using a measurable baseline between two observation points and the angles each point sights toward the target — the Law of Sines then recovers the unmeasurable distance.
Why does this work without measuring the actual distance?
Because a baseline and two angles fully determine a triangle's shape and size (the ASA congruence condition), and the Law of Sines relates every side to its opposite angle through one shared ratio — the target distance is simply the side this ratio can recover without ever measuring it directly.
What if the two angles sum to 180° or more?
No real triangle exists with that combination, since the third angle would be zero or negative — the two observation points would be sighting in directions that never actually converge on a single point.
Is this the same technique used in GPS or surveying equipment?
The core geometric idea is the same, though modern GPS relies on timing signals from multiple satellites (technically trilateration, using distances rather than angles) rather than sighted angles from a physical baseline — but classical land surveying still uses angle-based triangulation directly.
Does the baseline need to be a specific length?
No — any convenient, directly measurable length works, as long as both endpoints can sight the same distant target. Shorter baselines suit nearby targets; much longer baselines (even the distance between two observatories) suit astronomical-scale triangulation.