Height & Distance
Height और Distance — tower, seedhi, patang
Height & Distance
- Trigonometry Basics
- Height & Distance
Solve tower, ladder and shadow problems using angles of elevation/depression and 30-60-90 triangles.
🎯 Learning Objective
Solve tower, ladder and shadow problems using angles of elevation/depression and 30-60-90 triangles.
💡 Concept
- Angle of elevation: looking UP from the horizontal; depression: looking DOWN from the horizontal
- Angle of depression from the top = angle of elevation from the bottom (alternate angles)
- Main weapon: tan θ = height / horizontal distance
- 30°-60°-90° triangle → sides in ratio 1 : √3 : 2 (opposite to 30°, 60°, 90°)
- Use √3 ≈ 1.732 when a numerical answer is demanded
🧮 Key Formulas
tan θ = height / distance
>
30-60-90 sides → 1 : √3 : 2
>
tan 30° = 1/√3, tan 45° = 1, tan 60° = √3
>
√3 ≈ 1.732
✏️ Easy Example
Q. From a point 20 m away from a tower's base, the angle of elevation of its top is 45°. Find the height of the tower.
- tan 45° = h/20
- 1 = h/20
- h = 20
Answer: 20 m
🇮🇳 Real-Life Example
Measure a mobile tower without climbing: walk back until its top appears at 45° — your distance from the base IS its height.
📝 Exam-Level Example
Q. The angle of elevation of the top of a tower from a point 30 m away is 30°. Find the height of the tower.
- tan 30° = h/30
- h = 30 × (1/√3)
- = 30/√3 = 10√3
Answer: 10√3 m ≈ 17.32 m
📝 Exam-Level Example
Q. A ladder makes a 60° angle with the ground and its foot is 10 m from the wall. Find the length of the ladder.
- cos 60° = 10/L
- 1/2 = 10/L
- L = 20
Answer: 20 m
🪄 Memory Trick
Three-speed rule: at 45° height = distance; at 30° height = distance/√3; at 60° height = distance × √3.
⚠️ Common Mistakes
- ❌ Measuring the angle of depression from the vertical instead of the horizontal
- ❌ Using sin when the hypotenuse is not involved — tan connects height and ground distance
- ❌ Leaving 30/√3 unrationalised and missing it among the options
🏆 Exam Tips
- ✅ Draw the right triangle FIRST and mark the 90° between tower and ground
- ✅ Depression questions: transfer the angle to the bottom as elevation, then solve normally
📌 Summary
- Elevation up, depression down — both from horizontal
- tan θ = height/distance is the main tool
- 1 : √3 : 2 sides for 30-60-90
- 45° → height = distance