🏗️ Heights & Distances with Answers | Board Exam Ready
30
Questions
15
2-Mark Qs
15
3-Mark Qs
7
Years Covered
📋 Key Topics (As per CBSE 2025-26 Syllabus)
Angle of Elevation — observer looks UP
Angle of Depression — observer looks DOWN
Tower / Building — find height or distance
Shadow Problems — Sun's altitude
Two Triangles — building + tower combined
Moving Objects — boat, car, plane speed
✅
Unit Weightage: Trigonometry unit carries 12 marks (Ch 8 + Ch 9). Chapter 9 alone can give 3–5 marks. Expect 1 SA (3m) + 1 MCQ (1m) or 1 LA (5m) from Heights & Distances. ALWAYS draw a diagram first!
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Key Values to Memorise: tan 30° = 1/√3 | tan 45° = 1 | tan 60° = √3 | sin 30° = 1/2 | sin 60° = √3/2 | cos 30° = √3/2 | cos 60° = 1/2 | √3 ≈ 1.732
📝 SECTION A — 2-MARK QUESTIONS
15 Questions • Very Short Answer (VSA) Type • Board PYQs 2018–2025
Q1
CBSE 2025
From a point on the ground, which is 30 m away from the foot of a vertical tower, the angle of elevation of the top of the tower is found to be 60°. Find the height of the tower.
2 MarksTower Height⭐ MOST RECENT
▼
∴ Height of tower = 30√3 m ≈ 51.96 m
Q2
CBSE 2024
The angle of depression of a car standing on the ground, from the top of a 85 m high tower is 45°. Find the distance of the car from the base of the tower.
2 MarksAngle of Depression
▼
∴ Distance of car from base = 85 m
Q3
CBSE 2024
A pole casts a shadow of length 2√3 m on the ground, when the Sun's elevation is 60°. Find the height of the pole.
2 MarksShadow Problem
▼
∴ Height of pole = 6 m
Q4
CBSE 2023
The ratio of the height of a tower and the length of its shadow on the ground is √3 : 1. What is the angle of elevation of the Sun?
2 MarksFind Angle⭐ IMPORTANT
▼
∴ Angle of elevation of Sun = 60°
Q5
CBSE 2023
If the length of the shadow of a vertical pole is equal to its height, find the angle of elevation of the Sun.
2 MarksShadow = Height
▼
∴ Angle of elevation of Sun = 45°
Q6
CBSE 2022
A ladder 15 m long just reaches the top of a vertical wall. If the ladder makes an angle of 60° with the wall, find the height of the wall.
2 MarksLadder Problem
▼
∴ Height of wall = 7.5 m
Q7
CBSE 2022
The tops of two poles of heights 20 m and 14 m are connected by a wire. If the wire makes an angle of 30° with the horizontal, find the length of the wire.
2 MarksWire between Poles
▼
∴ Length of wire = 12 m
Q8
CBSE 2021
A kite is flying at a height of 30 m from the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is 60°. Find the length of the string (use √3 = 1.73).
2 MarksKite Problem
▼
∴ Length of string = 20√3 m ≈ 34.64 m
Q9
CBSE 2021
An observer 1.7 m tall is 20√3 m away from a tower. The angle of elevation from the eye of the observer to the top of the tower is 30°. Find the height of the tower.
2 MarksObserver Height Given
▼
∴ Height of tower = 21.7 m
Q10
CBSE 2020
From the top of a 25 m high tower, the angle of depression of a point on the ground is 30°. Find the distance of the point from the base of the tower.
2 MarksAngle of Depression
▼
∴ Distance = 25√3 m ≈ 43.3 m
Q11
CBSE 2020
A tree breaks due to a storm and the broken part bends so that the top of the tree touches the ground making an angle 30° with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.
2 MarksBroken Tree⭐ CLASSIC
▼
∴ Height of tree = 8√3 m ≈ 13.86 m
Q12
CBSE 2019
A ladder leaning against a wall makes an angle of 60° with the horizontal. If the foot of the ladder is 2.5 m away from the wall, find the length of the ladder.
2 MarksLadder + Wall
▼
∴ Length of ladder = 5 m
Q13
CBSE 2019
If the angle of depression of an object from a 45 m high temple is 30°, find the distance of the object from the temple.
2 MarksDepression from Temple
▼
∴ Distance = 45√3 m ≈ 77.94 m
Q14
CBSE 2018
In the figure, AB is a 6 m high pole and CD is a ladder inclined at an angle of 60° to the horizontal and reaches up to a point D of the pole. If AD = 2.54 m, find the length of the ladder. (Use √3 = 1.73)
2 MarksLadder to Pole
▼
∴ Length of ladder = 4 m
Q15
CBSE 2018
A tower AB is 20 m high and BC, its shadow on the ground, is 20√3 m long. Find the Sun's altitude (angle of elevation).
2 MarksFind Sun's Altitude
▼
∴ Sun's altitude = 30°
📝 SECTION B — 3-MARK QUESTIONS
15 Questions • Short Answer (SA) Type • Board PYQs 2018–2025
Q16
CBSE 2024
A man on a cliff observes a boat at an angle of depression of 30° which is approaching the shore with uniform speed. Six minutes later, the angle of depression is 60°. Find the time taken by the boat to reach the shore.
3 MarksMoving Boat + Time⭐ MOST RECENT
▼
∴ Time to reach shore = 3 minutes
Q17
CBSE 2025
From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 45°. Determine the height of the tower.
3 MarksBuilding + Tower⭐ MOST IMPORTANT
▼
∴ Height of cable tower = 7(1 + √3) m ≈ 19.12 m
Q18
CBSE 2023
A straight highway leads to the foot of a tower. A man standing on the top of the 75 m high tower observes two cars at angles of depression of 30° and 60°, which are approaching the tower. If one car is exactly behind the other on the same side, find the distance between the two cars. (Use √3 = 1.73)
3 MarksTwo Cars on Highway⭐ MOST ASKED TYPE
▼
∴ Distance between two cars = 50√3 m = 86.5 m
Q19
CBSE 2023
From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 30°. Determine the height of the tower.
3 MarksBuilding + Cable Tower
▼
∴ Height of cable tower = 28 m
Q20
CBSE 2022
The shadow of a tower standing on level ground is found to be 40 m longer when the Sun's altitude is 30° than when it is 60°. Find the height of the tower.
3 MarksShadow Difference⭐ CLASSIC
▼
∴ Height of tower = 20√3 m ≈ 34.64 m
Q21
CBSE 2021
A straight highway leads to the foot of a tower. A man standing on the top observes a car at angle of depression 30°, approaching with uniform speed. 6 seconds later, the angle of depression becomes 60°. Find the time taken by the car to reach the foot of the tower.
3 MarksCar Approaching Tower
▼
∴ Time to reach foot = 3 seconds
Q22
CBSE 2020
A statue 1.6 m tall stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60° and from the same point, the angle of elevation of the top of the pedestal is 45°. Find the height of the pedestal. (Use √3 = 1.73)
3 MarksStatue + Pedestal⭐ IMPORTANT
▼
∴ Height of pedestal = 0.8(√3+1) m ≈ 2.19 m
Q23
CBSE 2020
The angles of depression of the top and the bottom of an 8 m tall building from the top of a multi-storeyed building are 30° and 45° respectively. Find the height of the multi-storeyed building and the distance between the two buildings.
3 MarksTwo Angles of Depression⭐ HOTS
▼
∴ Height = 4(3+√3) ≈ 18.93 m; Distance = 4(3+√3) ≈ 18.93 m
Q24
CBSE 2019
As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships. (Use √3 = 1.73)
3 MarksTwo Ships + Lighthouse⭐ CLASSIC
▼
∴ Distance between ships = 75(√3–1) ≈ 54.75 m
Q25
CBSE 2019
A moving boat is observed from the top of a 150 m high cliff moving away from the cliff. The angle of depression changes from 60° to 45° in 2 minutes. Find the speed of the boat in m/hr.
3 MarksBoat Speed
▼
∴ Speed = 1500(3–√3) m/hr ≈ 1902 m/hr
Q26
CBSE 2018
As observed from the top of a 100 m high lighthouse, the angles of depression of two ships are 30° and 45°. If one ship is exactly behind the other on the same side, find the distance between the two ships. (Use √3 = 1.732)
3 MarksLighthouse + Ships
▼
∴ Distance between ships = 100(√3–1) = 73.2 m
Q27
CBSE 2022
A man on the deck of a ship, 12 m above water level, observes that the angle of elevation of the top of a cliff is 60° and the angle of depression of the base of the cliff is 30°. Find the distance of the cliff from the ship and the height of the cliff. (Use √3 = 1.73)
3 MarksShip + Cliff⭐ HOTS
▼
∴ Distance = 12√3 ≈ 20.78 m; Height of cliff = 48 m
Q28
CBSE 2021
A 1.2 m tall girl spots a balloon moving with the wind in a horizontal line at a height of 88.2 m from the ground. The angle of elevation of the balloon from the eyes of the girl at any instant is 60°. After 30 seconds, the angle of elevation reduces to 30°. Find the distance travelled by the balloon during this interval. (Use √3 = 1.73)
3 MarksBalloon Moving
▼
∴ Distance travelled = 58√3 m ≈ 100.34 m
Q29
CBSE 2018
The angle of elevation of the top of a building from the foot of a tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 50 m high, find the height of the building.
3 MarksTower + Building
▼
∴ Height of building = 50/3 m ≈ 16.67 m
Q30
CBSE 2018
A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from 30° to 60° as he walks towards the building. Find the distance he walked towards the building.
3 MarksBoy Walking
▼
∴ Distance walked = 19√3 m ≈ 32.91 m
📐 KEY FORMULAS & CONCEPTS
Core Trig Ratios (Right △):
tan θ = Opposite / Adjacent (MOST USED in this chapter)
sin θ = Opposite / Hypotenuse
cos θ = Adjacent / Hypotenuse
Standard Values:
tan 30° = 1/√3 ≈ 0.577 | tan 45° = 1 | tan 60° = √3 ≈ 1.732
sin 30° = 1/2 | sin 45° = 1/√2 | sin 60° = √3/2
cos 30° = √3/2 | cos 45° = 1/√2 | cos 60° = 1/2 Constants: √3 ≈ 1.732 | √2 ≈ 1.414
Key Concepts:
• Angle of Elevation → observer looks UP to object
• Angle of Depression → observer looks DOWN to object
• Angle of depression from top = angle of elevation from bottom (alternate angles)
• Speed = Distance / Time
• Distance between objects = |d₂ – d₁| (same side) or d₁ + d₂ (opposite sides)
💡 BOARD EXAM TIPS — HEIGHTS & DISTANCES
ALWAYS draw diagram first: Marks are given for the diagram itself (½ to 1 mark). Label all points, angles, heights, and distances clearly.
tan θ is your best friend: 90% of problems use tan θ = height/distance. Only use sin/cos when string/rope/hypotenuse is involved.
Alternate angles rule: Angle of depression from TOP = angle of elevation from BOTTOM (due to horizontal lines being parallel).
Two triangle problems: Find common side first (usually distance d), then use it in the second triangle to find the unknown.
Rationalise denominators: When answer has √3–1 in denominator, multiply by (√3+1)/(√3+1). Board expects simplified answers.
Expected marks: 3–5 marks from this chapter. Usually 1 SA/LA question + 1 MCQ. Case-study based question also possible.