Theodolite

A theodolite is an instrument that measures horizontal angles (azimuths) and vertical angles (elevations). Used in surveying and geodesy, it serves to determine heights and distances by triangulation, without direct access to the target object.

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How to measure it in class

The FizziQ theodolite measures an elevation angle thanks to the accelerometer, which detects the direction of the vertical, and an azimuth thanks to the magnetometer, which locates magnetic north.

Steps:

Open FizziQ and select the Theodolite tool. Lay the smartphone flat on a horizontal table and check that it displays 0° of elevation; if not, calibrate.
Stand at a marked spot and measure with a tape measure the horizontal distance d separating the observer’s feet from the base of the target object. It is this measurement that limits the precision of the final result.
Aim at the top of the object by aligning the edge of the smartphone with the line of sight, without moving your feet, and read the elevation angle α.
Measure the height of the observer’s eyes above the ground, to be added to the result.
Take three successive sightings and use the average: the spread between the measurements gives the actual uncertainty.
For an azimuth, point the smartphone toward the target and read the angle relative to north, keeping away from any metallic mass and any electrical appliance.

Scientific activities on this topic

Tree height with a theodolite: measure the height of an inaccessible tree from a ground distance and an elevation angle.
Law of sines with a theodolite: verify the law of sines experimentally in a triangle built in the field.
Triangulation with a theodolite: determine the position of an inaccessible point from two sightings and a known baseline.
Measuring the height of a building with a smartphone: video sequence from the La main à la pâte Foundation (in French).

Learn more

An instrument born from geodesy

The theodolite is a sighting instrument mounted on two perpendicular graduated circles: one measures horizontal angles (azimuths), the other vertical angles (elevations). Its modern form dates from the 16th century, but it becomes the central tool of geodesy in the 18th century, when Delambre and Méchain use it to measure the meridian arc that would serve to define the meter. Precision theodolites reach the arcsecond, that is 1/3600 of a degree.

What a smartphone really measures

A smartphone has neither a telescope nor a graduated circle: it reconstructs the two angles from its inertial sensors. The elevation angle comes from the accelerometer, which measures the direction of the gravity field and therefore provides a reliable vertical reference at rest. The azimuth comes from the magnetometer, which locates the Earth’s magnetic field. This difference of origin explains an important asymmetry: the elevation is measured very stably, whereas the azimuth is sensitive to magnetic disturbances.

The dominant sources of error

Contrary to intuition, the angle is almost never the limiting factor. The first source of error is the measurement of the horizontal distance on the ground: a miscounted pace, sloping ground or a poorly located tree base introduce several percent of deviation. The second is how the device is held: a trembling hand or a smartphone that is not exactly in the vertical plane of sight distorts the angle by several degrees. The third, specific to the azimuth, is the magnetic deviation due to metallic structures, concrete reinforcement, whiteboards or computers. Finally, the height of the observer’s eyes is often forgotten: it is 1.5 to 1.7 m, that is a systematic error of 10% on a 15 m tree.

Sensitivity of the tangent

The error made on the height depends strongly on the sighting angle. For α close to 45°, the tangent function varies slowly and a 1° error on the angle produces about 3% error on the height. For α = 70°, the same 1° error produces about 5.4%. You should therefore move away from the object until you sight at an angle between 30° and 50°, rather than standing at the foot of the object.

Orders of magnitude

Angular resolution of a smartphone in elevation: about 1°. In azimuth: 2 to 5° depending on disturbances. Optical site theodolite: 10 arcseconds. Precision geodetic theodolite: 0.5 arcsecond. Recommended sighting angle in class: 30 to 50°. Typical precision achieved on the height of a 15 m tree: 1 to 2 m, that is about 10%.

Formula

For an object seen at an elevation angle α from a horizontal distance d:

h = d × tan(α) + h_eye

where:

h: total height of the object (m)
d: horizontal distance between the observer and the foot of the object (m)
α: elevation angle measured with the theodolite (°)
h_eye: height of the observer’s eyes above the ground (m)

If the foot of the object is inaccessible, two sightings from two aligned points a distance b apart give:

h = b / (1/tan(α₂) − 1/tan(α₁))

where α₁ and α₂ are the elevation angles measured from the furthest and the nearest point.

The propagation of the uncertainty on the angle is written:

Δh / h ≈ Δα / (sin(α) × cos(α))

with Δα expressed in radians. This expression is minimal for α = 45°.

Application examples

A student 20 m from a tree sights its top at 35°. The height is 20 × tan(35°) + 1.6 = 14.0 + 1.6 = 15.6 m.
At 50 m from a building seen at 42°, the height reaches 50 × tan(42°) + 1.6 = 45.0 + 1.6 = 46.6 m, that is about fifteen floors.
An error of 1 m on a distance of 20 m translates directly into 5% error on the height: measuring the distance with a tape measure rather than counting paces changes everything.
In surveying, the theodolite combined with a laser rangefinder forms a total station, which gives angles and distances simultaneously and computes the coordinates of the sighted point.
Astronomers use the same principle under the name of altazimuth mount: altitude and azimuth are enough to point at a celestial body.
In navigation, the sextant is a cousin of the theodolite: it measures the height of a celestial body above the horizon to deduce the latitude.

FAQ

Q: Do you need to calibrate the smartphone theodolite before measuring? A: Yes for the azimuth: you have to trace a figure eight with the smartphone to recalibrate the magnetometer. For the elevation, it is enough to check that the device laid on a horizontal table indeed displays 0°.

Q: Why is my tree height measurement always too low? A: Almost always because the height of the eyes has been forgotten. The formula d × tan(α) gives the height above eye level, not above the ground: 1.5 to 1.7 m must be added.

Q: Can you measure an object whose foot is inaccessible, like a cliff on the other side of a river? A: Yes, by taking two sightings from two points aligned with the object and separated by a measured baseline b. The unknown distance is thus eliminated, at the cost of a larger uncertainty.

Q: Why is the azimuth less reliable than the elevation? A: Because the two angles do not come from the same sensor. The elevation relies on gravity, which is stable and the same everywhere. The azimuth relies on the Earth’s magnetic field, which is very weak and easily disturbed by surrounding metal.

Q: At what distance should you stand from the object? A: At a distance comparable to the height of the object, so as to sight at an angle of 30 to 50°. Too close, the angle approaches 80° and the tangent strongly amplifies the slightest sighting error.

Triangulation - Inclinometer - Accelerometer - Magnetometer - Orientation - Compass