Mathematics 11–12 · Year 11
Height of a tower from two ground stations: trigonometry in three dimensions
Further trigonometry: Trigonometry in three dimensions (Mathematics Extension 1, Year 11); Trigonometry and measure of angles: Trigonometry with acute angles (Mathematics Advanced, Year 11)
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
Two vertical right triangles sharing the tower and a horizontal triangle on the ground fit together in three dimensions, so an elevation from one station and the ground distances predict the elevation at the other.
What you need
- A phone clinometer (phyphox Inclination), a compass, a 100 m tape
- A tall object (a light tower, a flagpole or a building corner) on open level ground
How to do it
- Choose station A due south of the tower's base C and station B due east of it; tape AC and BC.
- Read the angle of elevation of the top from A (alpha) and from B (beta), adding the eye height in each calculation.
- Compute the height from A, then predict beta from tan(beta) = (height - eye height) / BC and compare with the reading.
- Compute AB from Pythagoras in the horizontal right triangle and check it against the tape.
- Draw the three-dimensional diagram with the two vertical triangles and the horizontal one, marking every measured value.
What you should see
Measured from eye level, AC = 40.0 m and alpha = 25.0 degrees give a rise of 18.7 m; from BC = 60.0 m the predicted beta is 17.3 degrees and AB = 72.1 m. The learner knows it worked when the beta predicted from station A matches the beta read at B within the clinometer's reading error.
What changes
- What you change
- station distances and the first elevation angle
- What you measure
- computed height and the predicted second angle
- What you keep the same
- level ground
- stations on perpendicular lines from the base
- eye height included at both stations
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- The two elevation angles can be added or averaged; they belong to different triangles with different bases.
- AB is the sum of AC and BC; the ground triangle is right-angled, so AB comes from Pythagoras.
Safety card
Hazards
- vehicles on open ground
Controls
- use a fenced field
- a partner watches while the observer sights
Note
No chemicals and no heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; outdoor work follows the school's grounds or excursion risk assessment (RiskAssess where the school uses it).
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 Extension 1 11–12 Syllabus (2024), Year 11 focus area Further trigonometry; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22ME1-11-03
- Mathematics Advanced 11–12 Syllabus (2024), Year 11 focus area Trigonometry and measure of angles; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MAV-11-04
- Australian Curriculum v9No Australian Curriculum v9 code is listed.
Sources
The pages the author read to write this activity.