Why do I need trigonometry to measure a tree?
A tree, its shadow, and a sunbeam make a right triangle. If you know how high the sun is in the sky and how long the shadow is, the tangent of that angle turns shadow length into height. Surveyors, builders, and hikers measure things they can't reach this way, without climbing anything.
- Height
- 4.37 m
- Sun ray (top to shadow tip)
- 12.8 m
Challenge: Find the angle where the shadow is exactly as long as the tree is tall.
Play
Make the sun lower or higher. Watch the shadow and the height change together.
Challenge: Find the angle where the shadow is exactly as long as the tree is tall. The box under the picture turns green when you get it.
Stuck? Pick one of the examples from the “Try an example” menu, or press “New example.”
Understand
The object, its shadow, and the sun's ray make a right triangle. The shadow is the side next to the angle (adjacent) and the height is the side across from it (opposite). Tangent links them:
When the sun is low, is small and shadows are long. At 45°,
and the shadow equals the height. As the sun climbs toward
90°, shoots up. Straight overhead it's undefined (#DIV/0!).
Use
Every input has a unit menu, so you can type values in the units you already have. Results follow your units.
Show the work
- Tangent: opposite over adjacent
\tan\theta = \frac{h}{s} ;\Rightarrow; h = s\,\tan\theta - With your numbers
h = 12 \times \tan 20^\circ = 12 \times 0.364 = 4.368\ \mathrm{m}
Export
Measure the shadow from the base of the object to the tip of the shadow. Get the sun's elevation from a sun-position app for your time and location, or use a clinometer.
- Measure on level ground. A slope makes the shadow longer or shorter.
- Do it quickly. The sun moves about 1° every 4 minutes.
For learning and estimation. Verify with applicable codes, standards, and a qualified professional before using in design, construction, or safety-critical work.
Cheat card
| Symbol | Meaning | Unit |
|---|---|---|
| height of the object | m | |
| shadow length | m | |
| sun’s angle above the horizon | ° |
- At 45°, the shadow is exactly as long as the object is tall.
- SOH-CAH-TOA: tangent is Opposite over Adjacent.
- No angle? Use your own shadow: tree height = tree shadow × (your height ÷ your shadow).
Where it’s used
- Civil & Construction
Surveyors find the heights of buildings and cliffs from an angle and a distance with the same tangent rule. - Hobbies & Crafts
Hikers, arborists, and photographers estimate heights of trees, towers, and waterfalls.
Related exhibits
Start hereLadder against a wall: how high will it reach?
A ladder, a wall, and the ground make a right triangle. Pythagoras does the rest.
Go furtherSpeed, distance, time: how long is the drive?
Distance is speed times time. Rearrange it and you know when you’ll arrive.
Also tryThrowing a ball: the quadratic arc
Every throw, jump, and fountain draws a parabola.
Questions people ask
How do I find a tree's height from its shadow?
Measure the shadow and find the sun's angle above the horizon. Multiply the shadow length by the tangent of that angle.
What if I don't know the sun's angle?
Measure your own shadow at the same moment. Your height and shadow make a similar triangle, so tree height = tree shadow × your height ÷ your shadow.
What happens when the sun is straight overhead?
There's no shadow, and tan 90° is undefined because it divides by cos 90° = 0. The site shows