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Trigonometric Height Estimation using a Handmade Clinometer

Experiment video

What you will learn

Construct an optical clinometer and determine the height of an inaccessible tall object (school building, tree, or flagpole) using the trigonometric tangent ratio (tan θ = opposite / adjacent).

Materials needed

180° Plastic Protractor with center hole
180° Plastic Protractor with center hole
₹50
What is it?
Surveyor's manual optical clinometer with weighted gravity pendulum and telescopic sight.
Specifications
Protractor Scale
180° semi-circular dial with 1° graduations
Sight Tube
150 mm brass optical sight with crosshair
Plumb Bob
Machined brass pointer bob on fine braided nylon line
Handle
Ergonomic wooden pistol grip
Straight drinking straw / hollow tube (optical sight)
Straight drinking straw / hollow tube (optical sight)
₹95
What is it?
Heavy-duty 30-meter fiberglass field surveyor's tape measure.
Specifications
Length
30.0 meters (100 feet)
Tape Material
Non-stretch vinyl coated fiberglass
Graduations
Millimeter and centimeter dual markings
Housing
Impact-resistant ABS case with rapid rewind crank
Fine cotton thread (25 cm) with metallic plumb weight / nut
Fine cotton thread (25 cm) with metallic plumb weight / nut
₹140
What is it?
Scientific calculator with trigonometric functions (tan, sin, cos).
Specifications
Precision
10-digit floating point arithmetic
Mode
DEG (Degree angle mode)
Heavy 30-meter Measuring Tape
Heavy 30-meter Measuring Tape
₹185
What is it?
Standard laboratory apparatus used in Trigonometric Height Estimation using a Handmade Clinometer.
Specifications
Category
Mathematics Apparatus
Grade
CBSE / NEP 2020 Laboratory Standard
Double-sided tape / craft glue
Double-sided tape / craft glue
₹230
What is it?
Standard laboratory apparatus used in Trigonometric Height Estimation using a Handmade Clinometer.
Specifications
Category
Mathematics Apparatus
Grade
CBSE / NEP 2020 Laboratory Standard
Scientific Calculator (for tan θ)
Scientific Calculator (for tan θ)
₹275
What is it?
Standard laboratory apparatus used in Trigonometric Height Estimation using a Handmade Clinometer.
Specifications
Category
Mathematics Apparatus
Grade
CBSE / NEP 2020 Laboratory Standard

Tick items as you collect them. Tap + to see the product, specifications and buy options.

Stay safe

Ensure the ground between the observer and the object is reasonably horizontal.
Wait until the plumb line stops swinging and hangs strictly vertical before recording angle α.
Keep straw sight aligned with the exact apex of the target object.
Include observer eye height (h1); omitting it introduces a 1.4-1.7 m systematic error.

Steps

0 of 8 steps done

1Affix the drinking straw firmly along the flat straight edge (0°-180° line) of the protractor using tape.
Observation
Construct a clinometer by attaching a clear drinking straw sight along straight edge of 180° protractor with plumb line bob suspended from center hole.
Safety
Ensure plumb line hangs freely without rubbing against protractor face.
2Tie one end of the fine thread through the center hole of the protractor and attach a metallic nut to the other end to act as a vertical plumb line.
Observation
Measure distance from observer stance to base of flagpole using 30-meter fiberglass surveyor's measuring tape (d = 10.0 m).
Safety
Keep measuring tape taut and horizontal along level ground.
3Measure the distance (d) from the observer's standing point to the base of the tall building/tree using the measuring tape.
Observation
Measure height of observer's eye from ground using measuring tape (h_eye = 1.50 m).
Safety
Stand with normal erect posture.
4Look through the straw sight to align the top of the object.
Observation
Sight flagpole peak through straw tube; clamp plumb string against protractor scale with finger when plumb bob settles.
Safety
Do not sight near the direct sun; sunlight through tube can cause optical injury.
5Hold the clinometer steady and have a partner read the angle (α) indicated by the plumb line on the curved scale.
Observation
Read angle on protractor: plumb line hangs at 52°, so elevation angle θ = 90° - 52° = 38.0°.
Safety
Double check: horizontal sight line gives 90° plumb reading (θ = 0°).
6Compute angle of elevation: θ = 90° - α.
Observation
Calculate height H = 1.50 + 10.0 × tan(38°) = 1.50 + 7.81 = 9.31 meters.
Safety
Repeat measurement from alternative distance d = 15.0 m to verify concordance.
7Measure the observer's eye height from ground level (h1).
8Calculate total object height: H = h1 + d · tan(θ).

Tap + on a step to see what you will observe and any safety tips.

Record your observations

Target Object Measured	Horizontal Distance d (m)	Elevation Angle θ (°)	tan(θ) Value	Calculated Height d·tan(θ) (m)	Observer Eye Level h (m)	Total Height H = h + d·tan(θ) (m)
School Flagpole	10.0 m	38.0°	0.7813	7.81 m	1.50 m	9.31 m
School Main Building Roof	15.0 m	35.0°	0.7002	10.50 m	1.50 m	12.00 m
Basketball Backboard Hoop	5.0 m	17.0°	0.3057	1.53 m	1.50 m	3.03 m (Official 3.05 m)
Campus Ashoka Tree Top	12.0 m	42.0°	0.9004	10.80 m	1.50 m	12.30 m
Light Pole Post	8.0 m	32.0°	0.6249	5.00 m	1.50 m	6.50 m

Tap a highlighted cell to write your own observation.

Result and conclusion

Measured elevation angle θ = 35° at distance d = 15.0 m with eye height h1 = 1.45 m. Computed height H = h1 + d · tan(35°) = 1.45 + 15.0 · (0.700) = 11.95 m (true height = 12.0 m).

Height H = h_eye + d · tan(θ) = 1.50 m + 10.0 m × tan(38.0°) = 1.50 m + 10.0 m × 0.7813 = 1.50 m + 7.81 m = 9.31 meters

Observation	What it means
Trigonometric Tangent Ratio	In right triangle formed by eye, tower top, and horizontal line, tan(θ) = Opposite / Adjacent = (H - h) / d.
Observer Eye Height Adjustment	Adding observer eye height h_eye = 1.50 m to calculated height d·tan(θ) corrects for ground clearance.
Basketball Hoop Benchmark	Calculated basketball hoop height of 3.03 m matches official NBA/FIBA standard 3.05 m within 0.7% error, validating method accuracy.
Distance Optimization	Positioning observer so elevation angle θ is around 30° to 45° minimizes angular reading error sensitivity on the clinometer dial.

Elevation Angle (θ) vs Calculated Target Height for Fixed d = 10m

Sample readings. Replace with your own results.

Experiment photos

Before the experimentDuring the experimentResult