Heights from shadows with similar triangles
Sun rays are parallel, so a stick and a building cast shadows in proportion.
Key formulas
Model
Two right triangles sharing the sun angle are similar. Heights over shadow lengths form equal ratios.
Example
Stick 1.5 m casts 2 m. Building casts 20 m. Building height = 1.5×20/2 = 15 m.
Assumptions
Ground is level and both objects stand vertical. A sloping lawn breaks the simple model.
Worked example
Worked example
Problem. A 1.5 m stick casts a 2 m shadow. At the same time a building casts a 20 m shadow. Find the building height.
Solution. Similar triangles: h/20 = 1.5/2 → h = 20 × 0.75 = 15 m.
Same time
Measure both shadows at the same time of day.
Review quiz
Person 1.7 m, shadow 1.2 m; tree shadow 9 m. Tree height is:
Practice questions
- Person 1.7 m, shadow 1.2 m; tree shadow 9 m. Find tree height.
- Why must the sun angle be the same for both?
Study notes
Is this trigonometry?
You can use tan, but similarity avoids naming the angle.