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Shadow Length

Shadow Length problems require calculating the length of a shadow cast by an object of known height, given the sun's angle of elevation. These problems use the trigonometric relationship: shadow length = height × cot(θ) = height / tan(θ), where θ is the sun's angle from the horizontal. They combine direction sense with basic trigonometry.

10Worksheets
200+Practice Questions
HardDifficulty
3-4 hoursHours to Master

What You'll Learn

Introduction & Overview Why This Matters How to Solve Pro Tips & Tricks
Shortcut Methods Common Mistakes Practice Worksheets Exam Importance

Introduction to Shadow Length

Shadow Length problems require calculating the length of a shadow cast by an object of known height, given the sun's angle of elevation. These problems use the trigonometric relationship: shadow length = height × cot(θ) = height / tan(θ), where θ is the sun's angle from the horizontal. They combine direction sense with basic trigonometry.

Prerequisites

Basic trigonometry (tan, cot) Sun angle concept (angle of elevation) Shadow direction knowledge Height measurement Formula: Shadow Length = Height / tan(θ)
Why This Matters: Shadow Length problems appear in 1-2 questions in advanced exams like CAT and Banking mains. They test integration of trigonometry with direction sense.

How to Solve Shadow Length Problems

1

Step 1: Identify the object's height (h) and sun's angle of elevation (θ)

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Step 2: Use formula: Shadow Length = h × cot(θ) = h / tan(θ)

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Step 3: For θ = 45°, shadow length = height (since tan45° = 1)

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Step 4: For θ < 45°, shadow length > height

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Step 5: For θ > 45°, shadow length < height

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Step 6: For θ = 90° (sun overhead), shadow length = 0

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Step 7: Express answer with appropriate units

Pro Strategy: Remember that tan(θ) = opposite/adjacent = height/shadow length. Therefore, shadow length = height / tan(θ). For common angles, memorize tan values: tan30°=1/√3≈0.577, tan45°=1, tan60°=√3≈1.732.

Example Problem

Example: A 10 m tall pole casts a shadow. The sun's angle of elevation is 30°. Find the shadow length (tan30° = 0.577). Solution: Step 1: Height = 10 m, θ = 30° Step 2: Shadow Length = 10 / tan30° = 10 / 0.577 ≈ 17.32 m Answer: Approximately 17.32 m

Pro Tips & Tricks

  • At sunrise/sunset (θ ≈ 0°): shadow length is very large (infinite at exact sunrise)
  • At noon (θ ≈ 90° in tropics, but in Northern Hemisphere, θ ≈ 90° - latitude) → shadow length minimum
  • When θ = 45°, shadow length = object height
  • When θ = 30°, shadow length ≈ 1.732 × height
  • When θ = 60°, shadow length ≈ 0.577 × height
  • Use approximate values when exact values aren't required

Shortcut Methods to Solve Faster

Shadow Length = Height × cot(θ)
cot(30°) = √3 ≈ 1.732, cot(45°) = 1, cot(60°) = 1/√3 ≈ 0.577
For θ = 30°, SL = H × 1.732
For θ = 45°, SL = H
For θ = 60°, SL = H × 0.577

Common Mistakes to Avoid

Using tan instead of cot (i.e., multiplying by tan instead of dividing)
Forgetting to use the correct trigonometric ratio
Not converting angle to degrees when using calculator
Confusing angle of elevation with angle from vertical
Using degrees instead of radians in calculations

Exam Importance

Shadow Length is an important topic for various competitive exams. Here's how frequently it appears:

SSC CGL
1-2 questions
BANKING PO
1-2 questions
RAILWAYS RRB
0-1 questions
CAT
1-2 questions
INSURANCE
1-2 questions

Ready to Master Shadow Length?

Start with Worksheet 1 and work your way up to expert level! Each worksheet includes:

20 practice questions
Detailed solutions
Step-by-step explanations
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