Measuring the height of a tree

This activity allows students to measure the height of inaccessible objects using trigonometry principles. It applies mathematical concepts to a concrete indirect measurement situation.

How tall is the tallest tree in your schoolyard? You could climb it with a tape measure, but mathematicians discovered a far more elegant solution over 2,600 years ago. According to legend, Thales of Miletus measured the height of the Great Pyramid of Egypt by comparing the length of its shadow with that of a stick of known height. The underlying principle, trigonometry, has been used ever since to measure everything from buildings to mountains to the distance to the Moon. The key insight is that if you know the distance to the base of an object and the angle at which you see its top, you can calculate its height using the tangent function: height = distance × tan(angle). Modern smartphones contain precise orientation sensors that can measure angles with accuracy better than one degree, effectively turning every phone into a digital theodolite. This experiment puts this ancient mathematical technique into practice, allowing students to measure the height of a tree, a building, or any tall structure without leaving the ground.

Learning objectives:

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.

Level:

Middle school

FizziQ

Author:

Duration (minutes) :

30

What students will do :

- Measure the angle of elevation to the top of a tall object using the FizziQ theodolite
- Apply the tangent formula to calculate the height of an inaccessible object
- Account for the observer's eye height in the calculation
- Evaluate measurement uncertainty and identify sources of error
- Connect the experiment to the historical and practical applications of trigonometry

Scientific concepts:

- Trigonometric tangent
- Elevation angle
- Theodolite
- Indirect measurement
- Triangulation

Sensors:

- Inclinometer / Theodolite (angle measurement using accelerometer and gyroscope)

What is required:

- Smartphone with the FizziQ application
- A large tree or building
- A tape measure to measure horizontal distance
- FizziQ experience notebook
- Calculator

Experimental procedure:

Choose a tall object to measure: a tree, flagpole, or building with a clearly visible top.
Stand at a point where you can see the top of the object clearly. Ensure there is a clear, level path between you and the base of the object.
Measure the horizontal distance from your position to the base of the object using a tape measure. Record this as d (in meters).
Open FizziQ and select the Theodolite tool (inclinometer mode).
Hold the smartphone at eye level and sight along the top edge toward the top of the object. When the crosshairs are aligned with the top, record the elevation angle α.
Measure your eye height h_eye (the height from the ground to your eyes) with a tape measure.
Calculate the height of the object: H = d × tan(α) + h_eye.
Repeat the measurement from a second distance (farther away or closer) and recalculate H. The two results should agree within measurement uncertainty.
Take the measurement from at least three different distances and calculate the average height and standard deviation.
If possible, verify your result by comparing with a known measurement (building plans, tree measurement by arborist, or shadow method).
Try measuring the angle from very close (steep angle) and very far (shallow angle). Discuss which distance gives the most accurate result.
Record all data in your FizziQ notebook: distance, angle, eye height, and calculated height for each trial.

Expected results:

For a typical school tree of 10-15 meters, the measured height should be accurate to within ±1-2 meters (5-15% relative error). The main sources of error are the angle measurement (±1-2° with a smartphone, translating to ±0.5-1.5 m at typical distances), the distance measurement (±0.1 m with a tape measure), and the identification of the exact top of the tree (canopy versus trunk tip). Measurements from very close range (angle > 60°) tend to be less accurate because the tangent function amplifies angular errors at steep angles. Measurements from very far away give shallow angles where small angle errors translate to large height errors. An optimal measurement distance is roughly equal to the object's height (giving an angle near 45°).

Scientific questions:

- Why is the optimal measurement distance approximately equal to the height of the object?
- How does the uncertainty in the angle measurement affect the calculated height?
- What was Thales' shadow method, and how does it relate to the tangent formula?
- Could you use this technique to measure the depth of a canyon or the height of a cliff from below?
- How do modern surveying instruments achieve greater accuracy than a smartphone?
- What are three real-world applications of trigonometric height measurement?

Scientific explanations:

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.

Extension activities:

Frequently asked questions:

Q: My two measurements from different distances give very different heights. What went wrong?
R: Check that you are measuring the horizontal distance to the base, not the slant distance. Also verify that the ground between you and the object is level. If the ground slopes, you need to correct for the elevation difference.

Q: The theodolite reading is unstable and keeps changing. How do I get a precise angle?
R: Hold the phone as steady as possible, ideally bracing your arms against your body. Take three readings and average them. Some phones have more stable orientation sensors than others.

Q: I am measuring a tree but cannot see the very top clearly. Does this matter?
R: Yes, aiming at a branch below the true top will underestimate the height. Try to sight the highest visible point and note that your result is a lower bound. Moving to a position where the top is clearly visible against the sky improves accuracy.

Q: Why do I need to add my eye height to the calculation?
R: The tangent formula gives the height above your eye level, not above the ground. Since you hold the phone at eye height, you must add the height of your eyes to get the total height of the object from the ground.