Mathematics 7–10 · Years 9–10

Height of a building with a clinometer: angle of elevation, distance and eye height

Measurement and space

Practical, model not builtLow risk

This site has no interactive model of its own. Where a step or a material names a Concept Studio model, simulation or tool, it has not been built; an external simulation a step names (for example PhET) is not part of this site.

The idea

The tangent ratio turns a measured angle of elevation and a measured horizontal distance into a height that cannot be measured directly, and the error in the answer depends on how well the angle was read.

Safety card

Low riskAn adult supervises

Hazards

walking backwards while sighting
sun exposure
roads or car parks near the object
looking towards the sun

Controls

partner watches the ground while the observer sights
choose an object with the sun behind the observer
stay on school grounds or the agreed area
hats and water

Note

No chemicals or heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; outdoor work follows the school's excursion and playground supervision rules.

What you need

Clinometer per group, built as on the NRICH page: a 180 degree protractor with a drinking straw taped along its straight edge, about 20 cm of string or strong cotton through the centre hole, and a small weight such as a metal nut, paper clips or a small piece of clay
30 m tape measure
Recording table: distance d (m), eye height (m), angle read (degrees), elevation θ (the difference between the reading and 90 degrees), computed height (m)
A tall object with a known height for calibration (a flagpole or a building whose plans give the height), and one whose height is unknown

How to do it

Measure the observer's eye height above the ground while standing.
Stand a measured distance d from the base of the object (20 m is workable) and sight the top through the straw while a partner reads where the thread crosses the protractor. The thread crosses the 90 degree mark when the straw is level, so the elevation is the difference between the reading and 90 degrees; check this first by sighting along something level.
Take three readings and use the median.
Compute height = d × tan θ + eye height. Repeat from a second distance (30 m) and compare the two heights.
Find the distance at which θ = 45 degrees; there the height above eye level equals the distance, a check with no calculator.
Compare with the known height for the calibration object and record the percentage error; then measure the unknown object.

What you should see

At d = 20.0 m, θ = 30 degrees and eye height 1.5 m the height is 20.0 × 0.5774 + 1.5 = 13.05 m. A 1 degree error in the angle at 30 degrees changes the height by d × sec²θ × (π/180) = 0.47 m, so a height reported to the nearest 0.5 m is honest from one reading; two distances agree within about 1 m when the angles were read to 1 degree. At the 45 degree spot the height above eye level equals d. The learner knows it worked when the calibration object's computed height is within 5 per cent of its known height and the two-distance results agree within that error.

What changes

What you change: distance from the base
What you measure: computed height (should stay constant)
What you keep the same: same observer and eye height
same clinometer
level ground
top sighted at the same point

Common misconceptions

Each of these ideas is wrong, and the activity is a chance to test it.

The angle read on the protractor is the angle of elevation (on a standard protractor it is the difference between the reading and 90 degrees).
Eye height can be ignored.
The tangent ratio changes with the size of the triangle.

Curriculum references

The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.

Mathematics K–10 Syllabus (2022), Stage 5 Trigonometry A; page read 2026-09-22MA5-TRG-C-01
Mathematics K–10 Syllabus (2022), Stage 5 Trigonometry B; page read 2026-09-22MA5-TRG-C-02
Australian Curriculum v9AC9M9M03AC9M10M03AC9M9SP01

Sources

The pages the author read to write this activity.

All Concept Studio activities