Measuring the height of a tree
Measure the height of a tree with a smartphone
Author:
Title 4
Learning objectives :
This activity allows students to measure the height of inaccessible objects using trigonometry principles. It applies mathematical concepts to a concrete indirect measurement situation.
Concepts covered
Trigonometric tangent; Elevation angle; Theodolite; Indirect measurement; Triangulation
What students will do :
The student uses the FizziQ theodolite to measure the angle of elevation of the top of a tree seen from a known position. By combining this measurement with the horizontal distance to the tree and the tangent formula the student can calculate the height of the tree without having to climb it or use other measuring instruments.
What is required :
Smartphone with the FizziQ application; A large tree or building; A tape measure to measure horizontal distance; FizziQ experience notebook; Calculator
Scientific background :
This method for measuring the height of inaccessible objects dates back to Antiquity and was already used by mathematicians like Thales of Miletus (6th century BC). It exploits the properties of right triangles and trigonometric ratios. The principle is simple: if we know the horizontal distance d to the object and the elevation angle α of its vertex, then its height h = d×tan(α). The FizziQ theodolite uses the smartphone's orientation sensors to measure this elevation angle with an accuracy of approximately ±1°. This measurement corresponds to the angle between the horizontal and the line of sight towards the top of the tree. An important subtlety: the height calculated by the formula corresponds only to the height difference between the observer's eye level and the top of the tree. To obtain the total height, add the height of the observer's eyes in relation to the ground (typically 1.5-1.7 m). The accuracy of this method depends on several factors: 1) The accuracy of the horizontal distance measurement; 2) The accuracy of the measured angle; 3) The verticality of the tree; 4) The flatness of the land. On flat ground, with a well-calibrated smartphone, the error is generally less than 5% for average-sized trees. This technique has applications in many fields: forestry, topography, astronomy (to estimate the height of celestial objects), and architecture. It perfectly illustrates how seemingly abstract mathematical concepts can solve practical problems, and is an excellent introduction to applied trigonometry.