Heights and distances: measuring a temple tower
Stand back from a temple tower, aim a clinometer at its top and use tan of the angle of elevation to find how tall it is.
Core concept
In a right-angled triangle, tan θ = opposite ÷ adjacent, so height = distance × tan(angle of elevation) + eye height
Topic: Trigonometry: heights and distances
Students want to know how tall a tiered temple tower is, without climbing it. One student stands some distance d from the tower and looks at the top through a clinometer, which measures the angle: The amount of turn between two lines that meet, measured in degrees (°). All key words of elevation θ above the horizontal. The eye level, the tower and the sight line make a right-angled triangle: the distance d is the side next to θ (adjacent) and the height above the eye is the side opposite θ. Since tan θ = opposite ÷ adjacent, the height above the eye is d × tan θ, and the student's eye height (1.5 m) is added to get the tower's height.
What you will learn
- Identify the opposite and adjacent sides for an angle of elevation.
- Use tan θ to find a height from a distance and an angle.
- Explain why different distances give different angles but the same height.
Before you start
- Right-angled triangles and the tangent ratio.
Materials
- A homemade clinometer (protractor, straw, thread and a weight)
Equipment
- A measuring tape
Procedure
- Drag the student along the ground to choose the distance from the tower.
- Drag the red sight line to aim it until it meets the top of the tower.
- Read the working: d × tan θ, plus the eye height.
- Find the height from 20 m, from 30 m using a 45° angle, and from 60 m, and record one sight line that passes above the top.
How this simulation is simplified
- The ground is level and the eye is 1.5 m above it. The tower is 31.5 m tall.