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CBSE · Class 10 · Mathematics

Applications of Trigonometry 9.1: Heights and Distances

5 concepts · 40 practice questions with answers · Free to read

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Angles of elevation and depression: meaning and basics

From daily life

Looking up at a kite, you tilt your head up from level; looking down from a balcony, you tilt it down. That tilt is the angle.

Simple definition

The line of sight joins the observer’s eye to the object. The angle of elevation is the angle between the horizontal and the line of sight when looking up; the angle of depression is the same angle when looking down. Both are measured from the horizontal.

θθelevationdepressionhd

In the figure: From the ground the angle of elevation is θ; from the top the angle of depression is also θ.

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Angles of elevation and depression: meaning and basics

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Rule 1: Name the angle — Ask: is the observer looking up or down from the horizontal?

  • Easy: Looking up at the top of a tower: angle of elevation.
  • Medium: Looking down at a boat from a cliff: angle of depression.
  • Exam-type: Both angles are acute and measured from the horizontal line.

Rule 2: Elevation equals depression — The two horizontals are parallel, so the alternate angles are equal.

  • Easy: Depression 40° from the top means elevation 40° from the bottom.
  • Medium: Always draw a horizontal line first, then mark the angle.
  • Exam-type: If the observer’s eye is 1.5 m above ground, add 1.5 m to the height found.
Remember

Angles of elevation and depression are always taken from the horizontal, never from the vertical.

Common mistake

Marking the angle of depression at the bottom of the figure; it belongs at the observer, below the horizontal there.

Memory clue

"Up from level is elevation; down from level is depression."

Practice questions

Easy
  1. 1From the top of a 24 m pole, the angle of depression of a car on the ground is 45°. What is the angle of elevation of the top of the pole from the car?
  2. 2From the top of a 76 m flagpole, the angle of depression of a car on the ground is 15°. What is the angle of elevation of the top of the flagpole from the car?
  3. 3From the top of a 74 m lighthouse, the angle of depression of a car on the ground is 35°. What is the angle of elevation of the top of the lighthouse from the car?
Medium
  1. 4From the top of a 70 m tower, the angle of depression of a car on the ground is 40°. What is the angle of elevation of the top of the tower from the car?
  2. 5From the top of a chimney, the angle of depression of a point on the ground is 55°. What angle does the line of sight make with the chimney?
  3. 6The angle of elevation of the top of a lighthouse from a point on the ground is 50°. What angle does the line of sight make with the lighthouse?
Challenging
  1. 7From the top of a tower, the angle of depression of a point on the ground is 45°. What angle does the line of sight make with the tower?
  2. 8The angle of elevation of the top of a building from a point on the ground is 45°. What angle does the line of sight make with the building?
Show answers (try them yourself first)
  1. 145
  2. 215
  3. 335
  4. 440
  5. 535
  6. 640
  7. 745
  8. 845

Finding a height

From daily life

Surveyors measure a tower’s height without climbing it: they stand at a known distance and read one angle.

Simple definition

The tower, the ground and the line of sight make a right triangle. With the distance d known and the angle of elevation θ, the height is given by tan θ = h ÷ d, so h = d tan θ.

Formula: h = d tan θ

ABCd = 30 mh = ?line of sight60°tan θ = h ÷ dh = 30 tan 60° = 30√3 m

In the figure: At 30 m with elevation 60°: h = 30 tan 60° = 30√3 ≈ 51.9 m.

Live figure

Finding a height

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Rule 1: Use tan θ — Multiply the distance by the tangent of the angle.

  • Easy: d = 30 m, θ = 60°: h = 30√3 ≈ 51.9 m.
  • Medium: d = 20 m, θ = 45°: h = 20 × 1 = 20 m.
  • Exam-type: d = 12 m, θ = 30°: h = 12 ÷ √3 = 4√3 ≈ 6.93 m.

Rule 2: Add the eye height — The triangle starts at the eye, so the ground part is extra.

  • Easy: Eye 1.5 m high, building 20 m away, elevation 30°: 20 ÷ √3 ≈ 11.55 m.
  • Medium: Building height = 11.55 + 1.5 ≈ 13.05 m.
  • Exam-type: Skipping the 1.5 m would give a height that is too small.
Remember

Draw the right triangle and mark d, h and θ before writing any ratio.

Common mistake

Using sin or cos here; the height and the ground distance are the two legs, so tan is the link.

Memory clue

"Height = distance × tan of the angle."

Practice questions

Easy
  1. 1The angle of elevation of the top of a lighthouse from a point on the ground 23√3 m from its foot is 30°. Find the height of the lighthouse (in m).
  2. 2The angle of elevation of the top of a flagpole from a point on the ground 17√3 m from its foot is 45°. Find the height of the flagpole (in m).
  3. 3The angle of elevation of the top of a lighthouse from a point on the ground 16 m from its foot is 45°. Find the height of the lighthouse (in m).
Medium
  1. 4The angle of elevation of the top of a building from a point on the ground 12√3 m from its foot is 60°. Find the height of the building (in m).
  2. 5A tower stands on top of a building. From a point on the ground 57 m from the foot of the building, the angles of elevation of the top of the building and the top of the tower are 30° and 45°. Find the height of the tower (in m).
  3. 6A tower stands on top of a building. From a point on the ground 13 m from the foot of the building, the angles of elevation of the top of the building and the top of the tower are 45° and 60°. Find the height of the tower (in m).
Challenging
  1. 7An observer whose eyes are 2 m above the ground stands 47 m from a lighthouse and sees its top at an angle of elevation of 60°. Find the height of the lighthouse (in m).
  2. 8An observer whose eyes are 2 m above the ground stands 21 m from a tree and sees its top at an angle of elevation of 30°. Find the height of the tree (in m).
Show answers (try them yourself first)
  1. 123
  2. 217√3
  3. 316
  4. 436
  5. 557 − 19√3
  6. 613√3 − 13
  7. 72 + 47√3
  8. 82 + 7√3

Finding a distance, shadow or ladder length

From daily life

A pole’s shadow shortens as the sun climbs. The angle of the sun links the pole’s height to the shadow’s length.

Simple definition

When the height h is known, the ground distance is d = h ÷ tan θ. For a sloping ladder or rope of length l, use sin θ = h ÷ l or cos θ = d ÷ l, so l = h ÷ sin θ.

Formula: d = h ÷ tan θ; l = h ÷ sin θ

ABCshadow 15√3pole 15sun raytan 30° = 15 ÷ ss = 15 ÷ tan 30° = 15√3 m

In the figure: Pole 15 m, sun at 30°: shadow = 15 ÷ tan 30° = 15√3 ≈ 25.98 m.

Live figure

Finding a distance, shadow or ladder length

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Rule 1: Distance or shadow — Divide the height by tan θ.

  • Easy: Pole 10 m, sun at 45°: shadow = 10 ÷ 1 = 10 m.
  • Medium: Pole 15 m, sun at 30°: shadow = 15√3 ≈ 25.98 m.
  • Exam-type: Tower 50√3 m, elevation 60°: distance = 50√3 ÷ √3 = 50 m.

Rule 2: Ladder or rope length — A sloping length is the hypotenuse, so use sin or cos.

  • Easy: Ladder reaching 12 m at 60°: l = 12 ÷ (√3/2) = 8√3 ≈ 13.86 m.
  • Medium: Foot of that ladder: l cos 60° = 8√3 × 1/2 = 4√3 ≈ 6.93 m from the wall.
  • Exam-type: Check: 12² + (4√3)² = 144 + 48 = 192 = (8√3)².
Remember

A shadow, a distance and a base are all horizontal; a ladder, a rope or a line of sight is the slant.

Common mistake

Using tan for a ladder length; the ladder is the hypotenuse, which needs sin or cos.

Memory clue

"Slant needs sin or cos; level distance needs tan."

Practice questions

Easy
  1. 1A chimney is 4√3 m high. The angle of elevation of its top from a point on the ground is 30°. How far is the point from the foot of the chimney (in m)?
  2. 2A building is 16√3 m high. The angle of elevation of its top from a point on the ground is 30°. How far is the point from the foot of the building (in m)?
  3. 3A chimney is 16√3 m high. The angle of elevation of its top from a point on the ground is 60°. How far is the point from the foot of the chimney (in m)?
Medium
  1. 4A building is 29√3 m high. The angle of elevation of its top from a point on the ground is 30°. How far is the point from the foot of the building (in m)?
  2. 5The shadow of a flagpole on level ground is 18√3 m long when the sun's altitude (angle of elevation) is 60°. Find the height of the flagpole (in m).
  3. 6A flagpole is 27 m high. Find the length of its shadow (in m) on level ground when the sun's altitude is 60°.
Challenging
  1. 7A chimney is 28 m high. Find the length of its shadow (in m) on level ground when the sun's altitude is 30°.
  2. 8A tower is 19√3 m high. Find the length of its shadow (in m) on level ground when the sun's altitude is 60°.
Show answers (try them yourself first)
  1. 112
  2. 248
  3. 316
  4. 487
  5. 554
  6. 69√3
  7. 728√3
  8. 819

Angles of depression

From daily life

Standing on a lighthouse, a guard sees a boat below the level line; the farther the boat, the smaller the downward tilt.

Simple definition

For a problem with an angle of depression, draw the horizontal from the observer, mark the angle below it, and then move it to the ground triangle using equal alternate angles. The vertical height is the side opposite that angle.

60° to foot30° to top60 mnot to scaletower

In the figure: From a 60 m building: depression 60° to the foot and 30° to the top of a tower. Tower = 60 − 20 = 40 m.

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Angles of depression

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Rule 1: One object — Depression θ at the top equals elevation θ at the object.

  • Easy: From a 50 m tower, depression 30° to a car: distance = 50 ÷ tan 30° = 50√3 ≈ 86.6 m.
  • Medium: From a 75 m lighthouse, depression 45° to a ship: distance = 75 m.
  • Exam-type: At 45° the height and the distance are equal.

Rule 2: Two angles from one place — Find the drop to each point and subtract.

  • Easy: Building 60 m: foot of tower at depression 60°, so distance = 60 ÷ √3 = 20√3 m.
  • Medium: Drop to the top at 30°: 20√3 × (1/√3) = 20 m, so the tower is 60 − 20 = 40 m.
  • Exam-type: Check: the bigger depression angle goes with the nearer, lower point.
Remember

A larger angle of depression means a point closer to the foot of the observer’s building.

Common mistake

Using the depression angle at the wrong corner of the triangle; shift it to the ground by alternate angles first.

Memory clue

"Depression above equals elevation below."

Practice questions

Easy
  1. 1From the top of a 38 m high tower, the angle of depression of a car on the ground is 45°. How far is the car from the foot of the tower (in m)?
  2. 2From the top of a flagpole, the angle of depression of a car standing 42 m from its foot is 30°. Find the height of the flagpole (in m).
  3. 3From the top of a pole, the angle of depression of a car standing 45 m from its foot is 30°. Find the height of the pole (in m).
Medium
  1. 4From the top of a 3√3 m high lighthouse, the angles of depression of two ships on the same side of it (in line with its foot) are 30° and 60°. Find the distance between the ships (in m).
  2. 5From the top of a 18√3 m high lighthouse, the angles of depression of two ships on opposite sides of it (in line with its foot) are 45° and 60°. Find the distance between the ships (in m).
  3. 6From the top of a 24√3 m high lighthouse, the angles of depression of two ships on the same side of it (in line with its foot) are 30° and 60°. Find the distance between the ships (in m).
Challenging
  1. 7From the top of a 60 m high building, the angle of elevation of the top of a tower is 30° and the angle of depression of its foot is 45°. Find the height of the tower (in m).
  2. 8From the top of a 24 m high building, the angle of elevation of the top of a tower is 60° and the angle of depression of its foot is 60°. Find the height of the tower (in m).
Show answers (try them yourself first)
  1. 138
  2. 214√3
  3. 315√3
  4. 46
  5. 518 + 18√3
  6. 648
  7. 760 + 20√3
  8. 848

Two observations and two objects

From daily life

Walk toward a tower and its top seems to rise higher in the sky. Two such angles are enough to find both the height and the walk.

Simple definition

Let the tower height be h. Each position gives a distance d = h ÷ tan θ. If the two positions are on the same side, the distances subtract; if on opposite sides, they add. This gives one equation in h.

30°60°20 mh

In the figure: Elevations 30° and 60°, 20 m apart: h√3 − h/√3 = 20, so h = 10√3 ≈ 17.3 m.

Rule 1: Two observations of one tower — Write both distances in terms of h and subtract.

  • Easy: Elevations 30° and 60°, moved 20 m closer: 2h/√3 = 20, so h = 10√3 ≈ 17.3 m.
  • Medium: Check: distances are 30 m and 10 m, and 30 − 10 = 20 m.
  • Exam-type: The nearer point always has the bigger angle.

Rule 2: Two objects and one tower — Opposite sides add the distances; the same side subtracts them.

  • Easy: Tower 30 m, opposite sides at 30° and 60°: 30√3 + 10√3 = 40√3 ≈ 69.3 m apart.
  • Medium: Tower 20 m, same side at 45° and 30°: 20√3 − 20 = 20(√3 − 1) ≈ 14.6 m apart.
  • Exam-type: Draw both positions on a straight line through the foot of the tower.
Remember

Two positions give two equations; the unknowns h and the extra distance are found together.

Common mistake

Adding the distances when both positions are on the same side; they must be subtracted.

Memory clue

"Same side subtract, opposite sides add."

Practice questions

Easy
  1. 1Two points A and B on opposite sides of a tree, in line with its foot, see its top at angles of elevation 45° and 45°. If AB = 10 m, find the height of the tree (in m).
  2. 2A 6 m high tower is seen from two points A and B on opposite sides of it, in line with its foot, at angles of elevation 45° and 45°. Find AB (in m).
  3. 3A 26 m high lighthouse is seen from two points A and B on opposite sides of it, in line with its foot, at angles of elevation 45° and 45°. Find AB (in m).
Medium
  1. 4Two vertical poles of heights 9 m and 9 + 11√3 m stand on level ground. From the top of the shorter pole the angle of elevation of the top of the taller pole is 45°. Find the distance between the poles (in m).
  2. 5Two points A and B on opposite sides of a tower, in line with its foot, see its top at angles of elevation 60° and 30°. If AB = 8 m, find the height of the tower (in m).
  3. 6Two vertical poles of heights 18 m and 30 m stand on level ground. From the top of the shorter pole the angle of elevation of the top of the taller pole is 30°. Find the distance between the poles (in m).
Challenging
  1. 7Two points A and B on opposite sides of a tower, in line with its foot, see its top at angles of elevation 45° and 60°. If AB = 24√3 m, find the height of the tower (in m).
  2. 8Two points on the same side of a tower and in line with its foot are 4 m and 144 m from the foot. The angles of elevation of the top of the tower from the two points are complementary. Find the height of the tower (in m).
Show answers (try them yourself first)
  1. 15
  2. 212
  3. 352
  4. 411√3
  5. 52√3
  6. 612√3
  7. 736√3 − 36
  8. 824