Heights and Distances
356. The Stretching Shadow · how sun angles reveal hidden heights
The relationship between height and shadow difference always holds by trigonometry.
As the sun's elevation changes, a fixed height casts shadows of different lengths. Using tan θ = height / shadow at two angles links the change in shadow length to the height by trigonometry.
What this lesson covers
Try to break it
Drag A up or down to change the tower's height h. The 45° shadow (length h) and the 30° shadow (length h·√3) both scale with h. The gap between the two shadow tips is exactly h·(√3 − 1) — a constant fraction of the height, independent of where you set h.
The proof, step by step
Prove that the change in shadow length follows from the change in the angle of the sun.
- In ΔABC, ∠ACB = 45° and ∠ABC = 90°. So tan 45° = AB/BC ⇒ 1 = AB/BC ⇒ BC = AB.
- In ΔABD, ∠ADB = 30° and ∠ABD = 90°. So tan 30° = AB/BD ⇒ 1/√3 = AB/BD ⇒ BD = AB√3.
- The shadow difference is DC = BD - BC. Substituting the expressions: DC = AB√3 - AB = AB(√3 - 1).
- Given DC = 10 m, we solve AB(√3 - 1) = 10 ⇒ AB = 10/(√3 - 1) ≈ 13.66 m.
Worked example
The length of the shadow of a tower standing on a level plane is found to be 10 m longer when the sun's altitude is 30° than when it was 45°. Find the height of the tower.
Let height be h. At 45°, shadow = h. At 30°, shadow = h√3. Difference = h(√3 - 1) = 10. So h = 10/(1.732 - 1) = 10/0.732 ≈ 13.66 m.
- 10 m
- 13.66 m — correct
- 15 m
- 20 m