Mathematics
Triangulation
Quick fact
The ancient Greek astronomer Eratosthenes used triangulation to estimate the Earth's circumference with remarkable accuracy over 2,000 years ago.
Why this is interesting
How can you find your exact location on a map without GPS? The answer lies in a simple geometric trick used for centuries: a single triangle can pinpoint a spot from miles away.
Read the full explanation
Understanding Triangulation
Imagine you stand at point A and see a distant mountain peak. You measure the angle between the peak and a known line, say north. Then you walk a known distance to point B—this is your baseline—and measure the angle to the peak again. With these two angles and the baseline, you can draw a triangle. The unknown point (the peak) sits at the third corner. By using basic geometry, you can compute the distance from either point to the peak. Triangulation turns the problem of finding an unknown location into solving a triangle with known parts. It works because triangles are rigid: given enough measurements, the shape is fixed.
A deeper explanation
Triangulation relies on the law of sines and the fact that the sum of angles in a triangle is 180°. With a known baseline (side) and two base angles, the other sides and distances are determined uniquely. In practice, surveyors use theodolites to measure angles with high precision. This method is fundamental to cartography: by creating a network of interlocking triangles (a triangulation network), large areas can be mapped accurately. Modern GPS inverts the process: satellites broadcast their positions, and a receiver measures the time delay to calculate distances, effectively performing trilateration (a close cousin using distances instead of angles). Triangulation’s elegance lies in transforming an unmeasurable distance into measurable angles, a principle that underpins everything from ancient navigation to satellite positioning.