Core ConceptRead this first — the foundation of the topic
CORE CONCEPT
When you look up at a tall building, the angle your line of sight makes with the horizontal ground is called the angle of elevation. When you look down from a height, it's called the angle of depression. These angles help us calculate heights and distances we cannot measure directly
KEY RULES
The angle of elevation from point A to point B equals the angle of depression from point B to point A. Always draw a right triangle and identify the opposite side, adjacent side, and hypotenuse clearly. The horizontal distance remains constant in most problems.
Formula BlockMemorise — at least one formula appears in every paper
• tan θ = Height/Base (most used)
• sin θ = Height/Hypotenuse
• cos θ = Base/Hypotenuse
• When angle changes from α to β: New height = Base × (tan β - tan α) + Original height
Exam PatternsWhat examiners ask — read before attempting PYQs
SSC CGL consistently asks 1-2 questions on this topic. Common scenarios include: tower/building height from given distance, finding distance when height is known, problems involving two angles of elevation, lighthouse/ship problems, and ladder-wall problems.
ShortcutsUse these to save 30–60 seconds per question
- ANGLE CHANGE METHOD: When moving closer or farther from an object, use the formula: h = d₁ × tan α = d₂ × tan β, where h is height, d is distance, and α, β are angles. This eliminates the need to calculate height separately.
Worked ExampleSolve this step-by-step before moving on
Effective height to calculate = h - 1.8m (since man has height)
3
Step 3
tan 30° = (h - 1.8)/150
4
Step 4
1/√3 = (h - 1.8)/150
5
Step 5
h - 1.8 = 150/√3 = 150/1.732 = 86.6m
6
Step 6
h = 86.6 + 1.8 = 88.4m
WORKED EXAMPLE 2: From a point on ground, a tree top's angle of elevation is 45°. Moving 20m closer, the angle becomes 60°. Find tree height.
1
Step 1
Let tree height = h, original distance = d
2
Step 2
From original position: tan 45° = h/d, so h = d
3
Step 3
From new position: tan 60° = h/(d-20)
4
Step 4
√3 = h/(d-20) = d/(d-20) [since h = d]
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Step 5
√3(d-20) = d
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Step 6
1.732d - 34.64 = d
7
Step 7
0.732d = 34.64, so d = 47.32m
8
Step 8
Tree height h = d = 47.32m
Exam TrapsCommon mistakes students make — avoid these
#1: Students forget to account for the observer's height. When a person observes something, always subtract the person's height from the total height calculated. Many students calculate the total vertical distance but forget the observer is not on the ground level.
ADDITIONAL SHORTCUTS: For 30-60-90 triangles, use ratio 1:√3:2.
For 45-45-90 triangles, use ratio 1:1:√2. When angle of elevation doubles, use the identity tan(2θ) = 2tan(θ)/(1-tan²θ). Remember that complementary angles have reciprocal trigonometric ratios.
Key Points to Remember
Angle of elevation = angle looking up; angle of depression = angle looking down
tan θ = Height/Base is the most frequently used formula in height-distance problems
Always subtract observer's height from total calculated height
Angle of elevation from A to B = Angle of depression from B to A
For 30° angle: tan 30° = 1/√3 = 0.577
For 45° angle: tan 45° = 1
For 60° angle: tan 60° = √3 = 1.732
In two-angle problems, use h = d₁ × tan α = d₂ × tan β shortcut
Draw clear diagrams marking height, base, and angles before solving
Horizontal distance remains same; only vertical measurements change with angle
Exam-Specific Tips
tan 30° = 1/√3 = 0.5774 (exact value)
tan 45° = 1 (exact value)
tan 60° = √3 = 1.732 (exact value)
sin 30° = 1/2 = 0.5
cos 30° = √3/2 = 0.866
sin 45° = cos 45° = 1/√2 = 0.707
sin 60° = √3/2 = 0.866
cos 60° = 1/2 = 0.5
Practice MCQs
Heights & Distances — Practice Questions
9graded MCQs · easy to hard · full solution & trap analysis
From the top of a 30-metre tall building, the angle of depression to a point on the ground is 45°. How far is the point from the base of the building?
Practice 2easy
A person standing 15√3 metres away from a tower observes the angle of elevation to the top as 30°. What is the height of the tower?
Practice 3easy
An observer on the ground sees the top of a tree at an angle of elevation of 60°. If the observer is 20 metres away from the base of the tree, what is the height of the tree?
Practice 4easy
From a point on the ground 40 metres away from the base of a building, the angle of elevation to the top is 45°. What is the height of the building?
Practice 5medium
A man observes the angle of depression to a point on the ground from the top of a 30-metre building as 45°. How far is the point from the base of the building?
Practice 6medium
From the top of a cliff 80 metres high, the angle of depression to a boat in the sea is 30°. How far is the boat from the base of the cliff?
Practice 7hard
A man standing at point A observes the angle of elevation to the top of a tower at 30°. He walks 40√3 metres towards the base of the tower and observes the angle of elevation to be 60°. If the man's eye level is 1.5 metres above ground, find the height of the tower (in metres).
Practice 8hard
From the top of a cliff 120 metres high, the angles of depression to two boats on the water are 30° and 60° respectively. Both boats are on the same side of the cliff. Find the distance between the two boats (in metres).
Practice 9hard
A person standing on the ground observes the angle of elevation to the top of a building as 45°. After walking 50 metres towards the building, the angle of elevation becomes 60°. If the person's eye level is at 1.6 metres, find the height of the building (in metres). [Use √3 ≈ 1.732]
60-Second Revision — Heights & Distances
Remember: Always draw diagram first and mark given angles and distances clearly
Formula: tan θ = Height/Base is the primary formula for 90% of problems
Trap: Don't forget to subtract observer's height from calculated total height
Shortcut: Use tan 30° = 0.577, tan 45° = 1, tan 60° = 1.732 for quick calculations
Pattern: Two-angle problems use h = d₁ × tan α = d₂ × tan β relationship
Check: Angle of elevation and depression are always measured from horizontal line
Quick tip: 30-60-90 triangle sides are in ratio 1:√3:2