Triangulation
Measure the distance between 2 points by triangulation
Author:
Title 4
Learning objectives :
This activity allows students to use triangulation to measure distances that are inaccessible in the field. It concretely applies trigonometry principles to real geometric situations.
Concepts covered
Triangulation; Law of sines; Triangle geometry; Azimuth; Mapping
What students will do :
The student uses the FizziQ theodolite to measure angles between several reference points in the playground. After having identified four points forming two triangles, the student systematically measures the angles from each vertex then uses the law of sines to progressively calculate all the unknown distances from a single directly measured distance.
What is required :
Smartphone with the FizziQ application; An open space with visible reference points; A tape measure to measure a reference distance; Drawing materials to make the diagram; Calculator; FizziQ experience notebook
Scientific background :
Triangulation is a fundamental technique in geodesy and topography that allows the precise position of a point to be determined by measuring angles from known positions. This method has been used for centuries to make accurate maps long before GPS systems were invented. The principle is based on the law of sines, which establishes that in any triangle, the lengths of the sides are proportional to the sines of the opposite angles: a/sin(A) = b/sin(B) = c/sin(C). Azimuth, measured by the FizziQ theodolite, is the horizontal angle between a direction and magnetic north. To obtain the internal angle of a triangle at a given vertex, we must calculate the difference between the azimuths of the two other vertices seen from this point. In this activity, the solution strategy consists of progressing methodically: 1) Measure an initial distance (AC) which will serve as a reference; 2) Calculate AD using the law of sines in the triangle ACD; 3) Calculate CD in the same way; 4) Determine the position of point H (projection of C onto AB); 5) Calculate the distances AH and BH; 6) Finally, deduce AB. This progressive approach makes it possible to construct a complete map from a minimum of direct measurements. The accuracy of this method depends mainly on the accuracy of the angular measurements. With a digital theodolite offering an accuracy of ±1°, the error on a calculated distance is typically 2-5% for well-conditioned triangles (without too acute angles). Historically, this technique has enabled major advances such as the first precise measurement of the earth's meridian by Delambre and Méchain (1792-1799), which led to the definition of the meter as a universal unit of length.