Heights and Distances
353. The Sun's Shadow Angle · how height and shadow reveal the sun's elevation
tan(θ) = height / shadow length.
For a vertical object and its shadow on level ground, the sun's angle of elevation θ satisfies tan θ = height / shadow length. A higher sun (larger θ) casts a shorter shadow.
What this lesson covers
Try to break it
Drag S closer to the tower and the shadow shortens — θ grows toward 90° (sun overhead). Drag S far away and the shadow stretches — θ drops toward 0° (sun on the horizon). When the shadow equals h·√3, θ = 30°; when it equals h, θ = 45°; when it equals h/√3, θ = 60°.
The proof, step by step
Prove that tan θ equals the height divided by the shadow length.
- Let the tower height TB = h.
- Given the shadow length BS = √3 h.
- In right ΔTBS, tan S = TB / BS = h / (√3 h) = 1 / √3.
- We know that tan 30° = 1 / √3.
- Therefore, the angle of elevation S = 30°.
Worked example
A pole 10 m high casts a shadow 10√3 m long. What is the angle of elevation of the sun?
tan θ = height/shadow = 10/(10√3) = 1/√3. Since tan 30° = 1/√3, θ = 30°.
- 30° — correct
- 45°
- 60°
- 90°