What is the angle of elevation?

The angle of elevation is the angle formed between the horizontal line of sight and the line of sight looking upward toward an object. If you look up at the top of a building from the ground, the angle your line of sight makes with the horizontal is the angle of elevation.

What is the angle of depression?

The angle of depression is the angle formed between the horizontal and the line of sight looking downward toward an object. If you stand on a tower and look down at a car on the road, the angle your line of sight makes with the horizontal is the angle of depression.

What is the relationship between angle of elevation and angle of depression?

When the observer is looking up at an object (angle of elevation) and the object is looking down at the observer (angle of depression), these two angles are equal — they are alternate interior angles formed by a transversal cutting two parallel horizontal lines.

Which trigonometric ratio is used most in height and distance problems?

tan θ = Perpendicular / Base = Height / Horizontal Distance is used in almost all height and distance problems because most problems involve the height (perpendicular) and the horizontal distance (base), not the slant distance (hypotenuse).

What are the standard trigonometric values to memorise?

sin 30°=1/2, cos 30°=√3/2, tan 30°=1/√3; sin 45°=1/√2, cos 45°=1/√2, tan 45°=1; sin 60°=√3/2, cos 60°=1/2, tan 60°=√3; sin 90°=1, cos 90°=0, tan 90°=undefined.

Height and Distance — Formulas, Shortcuts, and Worked Examples | TNPSC, SSC, Bank

Quick Concept Summary

Height and Distance problems apply basic trigonometry to find heights of buildings, towers, trees, or distances across rivers and valleys — without physically measuring them.

The two key angles:

Term	Meaning	Visual
Angle of Elevation	Angle formed when looking up at an object	Observer on ground, object above
Angle of Depression	Angle formed when looking down at an object	Observer on height, object below

Critical rule: Angle of elevation from A to B = Angle of depression from B to A (alternate angles).

Key Formulas and Trigonometric Values

Standard Trig Ratios

Angle	sin	cos	tan
0°	0	1	0
30°	1/2	√3/2	1/√3
45°	1/√2	1/√2	1
60°	√3/2	1/2	√3
90°	1	0	undefined

Memory trick for sin: 0, 1/2, 1/√2, √3/2, 1 → think √0/2, √1/2, √2/2, √3/2, √4/2

Applied Formulas

Situation	Formula
Find height when distance and angle known	H = d × tan θ
Find distance when height and angle known	d = H / tan θ
Find hypotenuse (slant distance)	Hypotenuse = H / sin θ
Two angles from same point	Use tan for each and set up equation

Worked Examples

Example 1: Basic Angle of Elevation

Problem:
A ladder leans against a wall. The foot of the ladder is 6 m away from the wall
and makes an angle of 60° with the ground. Find the height of the wall where
the ladder touches it.

Given:
  Horizontal distance = 6 m
  Angle of elevation = 60°
  Height = ?

Step 1: Use tan formula
  tan 60° = Height / Horizontal distance
  √3 = Height / 6
  Height = 6√3 ≈ 6 × 1.732 = 10.39 m

Answer: Height = 6√3 m ≈ 10.4 m

Example 2: Find Distance from Height

Problem:
The angle of elevation of the top of a tower from a point on the ground is 30°.
If the tower is 20 m tall, how far is the point from the base of the tower?

Given:
  Height of tower = 20 m
  Angle of elevation = 30°
  Distance = ?

Step 1: Use tan formula
  tan 30° = Height / Distance
  1/√3 = 20 / Distance
  Distance = 20√3 ≈ 20 × 1.732 = 34.64 m

Answer: Distance = 20√3 m ≈ 34.64 m

Example 3: Angle of Depression

Problem:
From the top of a building 50 m high, the angle of depression of a car on
the road is 45°. Find the distance of the car from the base of the building.

Given:
  Height of building = 50 m
  Angle of depression = 45°
  Distance = ?

Step 1: Angle of depression = Angle of elevation from car to top = 45°
  tan 45° = Height / Distance
  1 = 50 / Distance
  Distance = 50 m

Answer: Distance of car from base = 50 m

Example 4: Two Observations from Same Point

Problem:
A person stands 40 m from a tower. The angle of elevation to the top is 60°
and to a flag on the top is also noted. If the flagpole above the tower makes
the total angle 60°, find the height of the tower. (Classic form: From a point,
angles to bottom and top of a flagpole on a tower are 30° and 60° respectively.
Tower height = 50 m. Find flagpole height.)

Restated Problem:
From a point P on the ground, the angle of elevation to the top of a tower is 30°
and to the top of a flagpole on the tower is 60°. The height of the tower is 50 m.
Find the height of the flagpole.

Given:
  Angle to top of tower (A) = 30°
  Angle to top of flagpole (B) = 60°
  Tower height = 50 m
  Let distance from P to base = d
  Let flagpole height = h

Step 1: From angle to top of tower
  tan 30° = 50 / d
  1/√3 = 50 / d
  d = 50√3 m

Step 2: From angle to top of flagpole
  tan 60° = (50 + h) / d
  √3 = (50 + h) / (50√3)
  50 + h = 50√3 × √3 = 50 × 3 = 150
  h = 100 m

Answer: Height of flagpole = 100 m

Example 5: Shadow Problems

Problem:
The shadow of a tower when the angle of elevation of the sun is 60° is found
to be 30 m shorter than when it is 30°. Find the height of the tower.

Given:
  Let height of tower = h
  Let shadow length at 30° = x
  Shadow length at 60° = x - 30

Step 1: At 30° elevation
  tan 30° = h / x
  1/√3 = h / x
  x = h√3

Step 2: At 60° elevation
  tan 60° = h / (x - 30)
  √3 = h / (h√3 - 30)
  √3(h√3 - 30) = h
  3h - 30√3 = h
  2h = 30√3
  h = 15√3 ≈ 25.98 m

Answer: Height of tower = 15√3 m ≈ 26 m

Example 6: Two Towers / Objects

Problem:
Two poles of heights 6 m and 11 m stand on a plane. The cables connecting their
tops make an angle of 30° with the horizontal. Find the distance between the poles.

Given:
  Height of shorter pole = 6 m
  Height of taller pole = 11 m
  Difference in heights = 11 - 6 = 5 m
  Angle = 30°
  Distance between poles = ?

Step 1: Draw a horizontal line from top of shorter pole to the taller pole.
  The vertical difference = 5 m (this is the "height" in our right triangle)
  tan 30° = 5 / distance
  1/√3 = 5 / distance
  distance = 5√3 ≈ 8.66 m

Answer: Distance between poles = 5√3 m ≈ 8.66 m

Shortcuts and Tricks

Trick 1: The 45° Shortcut

When angle = 45°, tan 45° = 1, so Height = Distance always. Most MCQ options with equal height and distance hint at 45°.

Trick 2: The 30°–60° Pair

If angle = 30°: Distance = Height × √3 (distance is longer)
If angle = 60°: Distance = Height / √3 (distance is shorter)

Trick 3: Shadow Length

Shadow length = Height / tan(sun's elevation angle) At 45° sun, shadow = height (equal length) At 60° sun, shadow = height/√3 (shorter shadow — sun is high) At 30° sun, shadow = height × √3 (longer shadow — sun is low)

Trick 4: Two Angles from Same Base

If angles of elevation from a point to the top and bottom of an object on a tower are α and β:

Height of object = d(tan β − tan α) where d = horizontal distance

TNPSC Exam Pattern

Question Type	Frequency	Difficulty
Find height using angle of elevation	Common	Easy–Medium
Find distance using angle	Common	Easy–Medium
Angle of depression from a height	Common	Medium
Two observations — find flagpole/tower height	Occasional	Medium–Hard
Shadow length problems	Occasional	Medium

Exam tips:

Almost always involves 30°, 45°, or 60° — memorise these three perfectly
Draw a diagram first — it prevents sign errors
tan θ is the ratio you'll use 90% of the time
Watch for "angle of depression" — it equals the angle of elevation from below

Practice Questions

Q1. A tower is 100 m tall. Find the angle of elevation from a point 100 m away. Answer: tan θ = 100/100 = 1, so θ = 45°

Q2. The angle of elevation of the sun is 30°. A pole casts a shadow 20 m long. Find the height of the pole. Answer: H = 20 × tan 30° = 20/√3 = 20√3/3 ≈ 11.55 m

Q3. From the top of a cliff 80 m high, the angle of depression of a boat is 30°. How far is the boat from the cliff base? Answer: d = 80/tan 30° = 80√3 ≈ 138.56 m

Q4. A man on the top of a tower sees two cars at angles of depression of 30° and 45° on opposite sides. Tower height = 100 m. Find total distance between cars. Answer: Distance to car 1 = 100/tan 45° = 100 m. Distance to car 2 = 100/tan 30° = 100√3 m. Total = 100 + 100√3 = 100(1 + √3) ≈ 273.2 m

Q5. From a point P, the angle of elevation to a tower top is 60°. From a point 20 m farther, it is 30°. Find the height of the tower. Answer: Let h = height, x = initial distance. tan 60° = h/x → h = x√3. tan 30° = h/(x+20) → h = (x+20)/√3. So x√3 = (x+20)/√3 → 3x = x+20 → x = 10. h = 10√3 ≈ 17.32 m

Q6. Two buildings face each other. From the roof of the shorter (8 m), the angle of elevation to the top of the taller is 60° and angle of depression to its base is 30°. Find the height of the taller building. Answer: Let d = distance between. tan 30° = 8/d → d = 8√3. tan 60° = extra height/d → extra = d√3 = 8√3 × √3 = 24 m. Total height = 8 + 24 = 32 m