7 min readConcept sheet1-mark MCQ or assertion-reason; a 2-mark definition or one-step height question

Line of sight, angle of elevation and angle of depression

The line of sight joins your eye to the object. Looking up, it makes an angle of elevation with the horizontal; looking down, an angle of depression.

Line of sightThe line from the observer's eye to the object being viewed

Angle of elevationAngle between the horizontal and the line of sight, when the object is above eye level

Angle of depressionAngle between the horizontal and the line of sight, when the object is below eye level

Key factDepression from A to B = elevation from B to A (alternate angles)

Angle of elevation vs angle of depression

Angle of elevationThe object is above the horizontal level of the eye, so you raise your head. Example: looking up at the top of the Qutub Minar from the ground.

Angle of depressionThe object is below the horizontal level of the eye, so you lower your head. Example: looking down from a lighthouse at a fishing boat.

Both angles are always measured from the horizontal line through the eye, never from the vertical.

Why the two angles are equal

Suppose you stand on a cliff top A and see a boat B below. Draw a horizontal line through A. The sea level through B is also horizontal, so the two lines are parallel. The line of sight AB cuts both lines. So the angle of depression at A equals the angle of elevation at B, as alternate interior angles. In your figure, always mark this angle inside the triangle at B. It saves many mistakes.

How to draw a heights and distances figure

1
Draw the ground as a horizontal line.
2
Draw the tower, tree or building as a vertical line from the ground. Mark the right angle at its foot.
3
Mark the observer's point on the ground, or at the top of a building.
4
Join the eye to the object. This is the line of sight.
5★
For a depression, draw a dotted horizontal line through the eye and mark the angle below it. Then mark the equal angle inside the triangle.
6
Write the given angle and lengths. Call the unknown h or x. If the observer's eye height is given, draw a horizontal line at eye level.

★ key step

Using the equal angles

Problem

A boy on the roof of a 12 m tall building sees a car on the road at an angle of depression of 30°. Find the angle of elevation of the boy from the car, and the distance of the car from the building. (Use √3 = 1.732)

Working

Let AB = 12 m be the building and C the car. ∠B = 90°.
Angle of elevation of A from C = angle of depression = 30° (alternate angles).
In ΔABC, tan 30° = AB/BC.
1/√3 = 12/BC, so BC = 12√3 m.
BC = 12 × 1.732 = 20.784 m ≈ 20.78 m.

Answer

Angle of elevation = 30°; the car is about 20.78 m from the building.

Adding the observer's height

Problem

A girl whose eyes are 1.4 m above the ground stands 25 m from a tree. The angle of elevation of the top of the tree from her eyes is 45°. Find the height of the tree.

Working

Draw a horizontal line from her eyes E to the tree, meeting it at D. ED = 25 m.
Let T be the top of the tree. In ΔEDT, tan 45° = TD/ED.
1 = TD/25, so TD = 25 m.
D is 1.4 m above the ground, the same as her eyes.
Height of the tree = TD + 1.4 = 25 + 1.4 = 26.4 m.

Answer

26.4 m

In real life

Coast guard staff at a lighthouse in Kanyakumari judge how far a ship is from its angle of depression. Builders find the height of a temple gopuram from an angle of elevation, without climbing it.

Exam angle

1 marks: “The angle of depression of a car from the top of a tower is 45°. What is the angle of elevation of the top of the tower from the car?” 45°, because the two angles are alternate interior angles between parallel horizontal lines.

2 marks: “From a point 40 m from the foot of a tower on level ground, the angle of elevation of its top is 60°. Find the height of the tower. (Use √3 = 1.732)” tan 60° = h/40, so h = 40√3 = 40 × 1.732 = 69.28 m.

3 marks: “A man whose eyes are 1.6 m above the ground stands 18 m from a pole. The angle of elevation of the top of the pole from his eyes is 30°. Find the height of the pole. (Use √3 = 1.732)” Height above eye level = 18 tan 30° = 18/√3 = 6√3 = 10.392 m. Pole = 10.392 + 1.6 = 11.992 ≈ 11.99 m.

Remember it

Elevation: eyes up. Depression: eyes down. Both start from the horizontal.