Angles of Elevation and Depression

Draw the horizontal reference line first — always. The angle opens between that horizontal and your line of sight, never from the vertical. Get the diagram right and the trigonometry is straightforward.

50–55 min MS-M2 — HIGH 3 MC 3 SA Lesson 16 of 22 Free
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Think First

You are standing on top of a cliff looking down at a boat. Your friend is in the boat looking up at you. You both measure the angle between the horizontal and your line of sight to each other. Will you get the same angle or different angles? Why?

Type your initial response below — you will revisit this at the end of the lesson.

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Key Relationships — This Lesson

$\theta_{\text{elev}} = \theta_{\text{dep}}$
Equal angles — the angle of elevation from A to B equals the angle of depression from B to A Reason: alternate interior angles between parallel horizontal lines cut by the line of sight
$\tan\theta = \dfrac{O}{A}$
Most common ratio — in elevation/depression problems: O = vertical height, A = horizontal distance Use $\sin$ or $\cos$ when the hypotenuse (line-of-sight distance) is involved
$h = d_1\tan\theta_1 = d_2\tan\theta_2$
Two-observation setup — both expressions equal the same height $h$; set equal and solve for $d_1$
ANGLE OF ELEVATION horizontal θ object observer θ measured UP from horizontal ANGLE OF DEPRESSION horizontal θ object observer θ measured DOWN from horizontal

🧠 Know

  • Angle of elevation: measured upward from horizontal
  • Angle of depression: measured downward from horizontal
  • Both angles are always referenced from the horizontal
  • Elevation angle from A to B = depression angle from B to A

💡 Understand

  • Why the horizontal reference line must appear in every diagram
  • Why elevation and depression angles are equal (alternate angles)
  • How two-observation problems share a common height

✅ Can Do

  • Draw a correct labelled diagram from any word description
  • Solve elevation and depression problems using SOHCAHTOA
  • Apply the equal-angle relationship to simplify depression problems
  • Set up and solve two-observation simultaneous problems
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Key Terms

Angle of elevation The angle measured upward from the horizontal to a line of sight — looking up at an object above you
Angle of depression The angle measured downward from the horizontal to a line of sight — looking down at an object below you
Horizontal reference line A horizontal line drawn from the observer's eye level — the baseline from which elevation and depression angles are measured; must appear in every diagram
Alternate interior angles Equal angles formed on opposite sides of a transversal cutting two parallel lines — the geometric reason elevation and depression angles are equal

Misconceptions to Fix

✗

Wrong: Angle of elevation from A to B equals angle of depression from B to A in all cases.

✗

Right: These angles are only equal when the horizontal lines at A and B are parallel. In problems with slopes or different heights, you must calculate each separately.

Angle of Elevation

Key Point

Always check your units before substituting into formulas. Converting to consistent units is a common source of errors in assessment tasks.

Key Terms
Youstanding on top of a cliff looking down at a boat
Your friendin the boat looking up at you
Both anglesalways referenced from the horizontal
elevation and depression anglesequal (alternate angles)
Area and perimetercalculated using the same formula
Converting to consistent unitsa common source of errors in assessment tasks

Looking Up — Angle of Elevation

The angle of elevation is measured from the horizontal upward to the line of sight. It is never measured from the ground if the ground is sloped, and never from the vertical.

Elements of a correct elevation

  • Observer at one end; horizontal reference line extending from observer
  • Line of sight goes upward from observer to the object
  • Angle of elevation sits between the horizontal and the line of sight
  • Right angle at the foot of the vertical height of the object
Common diagram error: Placing the right angle at the observer's position, or marking the angle from the vertical side. The right angle is at the base of the object's height — not at the observer.
Worked Example 1 Basic Elevation

Problem

From a point 45 m from the base of a vertical building, the angle of elevation to the top is 38°. Find the height of the building, correct to 2 decimal places.

Solution

1 Horizontal distance $= 45$ m (A); height $= x$ (O); $\theta = 38°$. Use $\tan\theta = \dfrac{O}{A}$ No hypotenuse involved; O and A relate to the angle → tangent

Angle of Depression

Looking Down — Angle of Depression

The angle of depression is measured from the horizontal downward to the line of sight. The observer is at the top; the horizontal reference line is at the observer's height, not at sea level or ground level.

Equal angles relationship: The angle of depression from B down to A equals the angle of elevation from A up to B. Both equal $\theta$. This works because the two horizontal lines (at A and at B) are parallel, and the line of sight is a transversal — the angles are alternate interior angles.

In practice: redraw the depression problem using the elevation angle at the base. You get the same triangle, the same answer — and the diagram is less prone to errors.

Worked Example 2 Angle of Depression

Problem

From the top of a cliff 80 m high, a boat is observed at sea with an angle of depression of 24°. Find the horizontal distance from the base of the cliff to the boat, correct to 2 decimal places.

Solution

1 Angle of depression $= 24°$ → angle of elevation from boat $= 24°$ (equal alternate angles) Redraw from base: O = 80 m (cliff height), A = $x$ (horizontal distance), $\theta = 24°$

Multi-Step and Finding Angles

Worked Example 3 Combined Heights

Problem

A 12 m antenna sits on top of a 40 m building. An observer stands 75 m from the base of the building. Find the angle of elevation to the top of the antenna, to the nearest minute.

Solution

1 Total height $= 40 + 12 = 52$ m (O); horizontal distance $= 75$ m (A) The observer is at ground level; the total height from ground to antenna top is the opposite side

Two-Observation Problems

Two Angles to the Same Object

When two angles of elevation to the same object are given from different points, you have two expressions for the same height. Setting them equal produces an equation solvable for the unknown distance.

Strategy:

  1. Let height $= h$, distance from nearer point to base $= d$
  2. Write $h = d \cdot \tan\theta_1$ (nearer point — larger angle)
  3. Write $h = (d + k) \cdot \tan\theta_2$ (further point — smaller angle, $k$ = separation)
  4. Set equal and solve for $d$
  5. Substitute back to find $h$
Check: The nearer point always has the larger angle of elevation. If your algebra gives a negative distance, you've labelled the points the wrong way around.
Worked Example 4 Two-Observation

Problem

From point A, the angle of elevation to the top of a tower is 52°. From point B, 30 m further from the tower on the same side, the angle of elevation is 38°. Find the height of the tower, correct to 2 decimal places.

Solution

1 Let $d$ = distance from A to tower base; height $= h$.
From A: $h = d\tan 52°$   … (1)
From B: $h = (d+30)\tan 38°$   … (2) Two triangles sharing the same height $h$; A is nearer (larger angle)
Practice

Practice Questions

For every question: draw a complete labelled diagram first (horizontal reference line, angle, right angle, all known measurements). Do not skip the diagram.

Section A — Basic Elevation and Depression

  1. From a point 60 m from the base of a vertical tree, the angle of elevation to the top is 42°. Find the height of the tree to 2 decimal places.
  2. From the top of a 120 m cliff, the angle of depression to a boat is 31°. Find the horizontal distance from the cliff base to the boat to 2 decimal places.
  3. A 15 m flagpole stands on top of a 25 m building. An observer is 80 m from the base. Find the angle of elevation to the top of the flagpole, to the nearest minute.
  4. From a lighthouse 65 m above sea level, a ship is observed with an angle of depression of 18°. Find the straight-line distance (line of sight) from the lighthouse to the ship, to 2 decimal places.

Section B — Finding Angles

  1. An observer on the ground is 50 m from a building. The top of the building is 35 m above the observer's eye level. Find the angle of elevation to the nearest minute.
  2. From the top of a 90 m tower, an observer looks down at a car on the road below. The car is 140 m from the base of the tower. Find the angle of depression to the nearest minute.

Section C — Two-Observation Problems

  1. From point P, the angle of elevation to the top of a tower is 45°. From point Q, 20 m further from the tower than P, the angle of elevation is 30°. Find the height of the tower to 2 decimal places.
  2. Two observers on opposite sides of a vertical cliff: Observer A is 80 m from the base and measures the angle of elevation as 55°; Observer B measures an angle of elevation of 40° from the other side. Find the height of the cliff and the total horizontal distance between the two observers.

Q1

$h = 60\tan42° \approx \mathbf{54.05 \text{ m}}$

Q2

$d = 120/\tan31° \approx 120/0.60086 \approx \mathbf{199.71 \text{ m}}$

Q3

Total height $= 40$ m; $\tan\theta = 40/80 = 0.5$; $\theta = \tan^{-1}(0.5) \approx 26.565° \approx \mathbf{26°34'}$

Q4

Line of sight $= H$; $\sin18° = 65/H$; $H = 65/\sin18° \approx 65/0.30902 \approx \mathbf{210.34 \text{ m}}$

Q5

$\tan\theta = 35/50 = 0.7$; $\theta = \tan^{-1}(0.7) \approx 34.99° \approx \mathbf{34°59'}$

Q6

$\tan\theta = 90/140$; $\theta = \tan^{-1}(9/14) \approx 32.73° \approx \mathbf{32°44'}$

Q7

From P: $h = d\tan45° = d$; From Q: $h = (d+20)\tan30°$; $d = (d+20)/\sqrt{3}$; $d\sqrt{3} = d+20$; $d(\sqrt{3}-1) = 20$; $d = 20/(\sqrt{3}-1) \approx 27.32$ m; $h = 27.32 \approx \mathbf{27.32 \text{ m}}$

Q8

$h = 80\tan55° \approx 114.23$ m; A side: $80$ m; B side: $h/\tan40° = 114.23/0.839 \approx 136.15$ m; Total $\approx \mathbf{216.15 \text{ m}}$

Revisit Your Initial Thinking

Look back at what you wrote in the Think First section. What has changed? What did you get right? What surprised you?

Multiple Choice

1 From a point 30 m from the base of a vertical wall, the angle of elevation to the top is 56°. The height of the wall is closest to:

A   20.14 m
B   24.86 m
C   44.47 m
D   53.96 m

? Regarding this topic, 1 From a point 30 m from the base of a vertical wall, the angle of elevation to the top is 56°. The height of the wall is closest to:

A     20.14 m
B     24.86 m
C     44.47 m
D     53.96 m
C - Correct!
C — $h = 30\tan56° = 30 \times 1.48256... \approx 44.47$ m.

2 From the top of a 50 m building, the angle of depression to a car on the street is 32°. The horizontal distance from the base of the building to the car is closest to:

A   26.49 m
B   31.25 m
C   59.98 m
D   80.02 m

? Regarding this topic, 2 From the top of a 50 m building, the angle of depression to a car on the street is 32°. The horizontal distance from the base of the building to the car is closest to:

A     26.49 m
B     31.25 m
C     59.98 m
D     80.02 m
D - Correct!
D — Using equal angles: elevation from car $= 32°$; $d = 50/\tan32° \approx 50/0.62487 \approx 80.02$ m.

3 An observer at ground level is 100 m from the base of a tower. The angle of elevation to the top is 48°. A flag on top adds 5 m to the height. The angle of elevation to the top of the flag, to the nearest degree, is:

A   48°
B   49°
C   50°
D   53°

? Regarding this topic, 3 An observer at ground level is 100 m from the base of a tower. The angle of elevation to the top is 48°. A flag on top adds 5 m to the height. The angle of elevation to the top of the flag, to the nearest degree, is:

A     48°
B     49°
C     50°
D     53°
B - Correct!
B — Tower height $= 100\tan48° \approx 111.06$ m; total $= 116.06$ m; new angle $= \tan^{-1}(116.06/100) \approx 49.25° \approx 49°$.

Short Answer

01

SA 4 3 marks From a point on level ground, the angle of elevation to the top of a vertical cliff is 34°. The point is 220 m from the base of the cliff.

(a) Draw a fully labelled diagram of this situation, including the horizontal reference line, angle, right angle, and all known measurements.  (1 mark)

(b) Find the height of the cliff correct to 2 decimal places.  (2 marks)

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(a)

Horizontal ground; observer at left with 34° angle of elevation marked; horizontal reference line from observer; line of sight to top of cliff; cliff vertical with right angle at base; 220 m horizontal distance and height $x$ labelled

(b)

$\tan34° = x/220$; $x = 220\tan34° = 220 \times 0.67450... = 148.39...\ \approx \mathbf{148.39 \text{ m}}$

02

SA 5 3 marks A ship is sailing toward a lighthouse. The lighthouse is 45 m tall, standing at the edge of a cliff at sea level. From the ship, the angle of elevation to the top of the lighthouse is 12°.

(a) Find the horizontal distance from the ship to the base of the lighthouse, to the nearest metre.  (2 marks)

(b) After the ship travels closer, the angle of elevation becomes 28°. How far has the ship travelled? Give your answer to the nearest metre.  (1 mark)

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(a)

$d_1 = 45/\tan12° \approx 45/0.21256 \approx 211.73 \approx \mathbf{212 \text{ m}}$

(b)

$d_2 = 45/\tan28° \approx 45/0.53171 \approx 84.64 \approx 85$ m; Distance travelled $= 212 - 85 = \mathbf{127 \text{ m}}$

03

SA 6 4 marks From point A on flat ground, the angle of elevation to the top of a vertical tower is 55°. From point B, which is 40 m further from the tower than A and on the same side, the angle of elevation is 35°.

(a) Let the horizontal distance from A to the base of the tower be $d$ metres and the height be $h$ metres. Write two equations for $h$ in terms of $d$.  (1 mark)

(b) By setting the two equations equal, find the value of $d$ correct to 2 decimal places.  (2 marks)

(c) Hence find the height of the tower correct to 2 decimal places.  (1 mark)

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(a)

From A: $h = d\tan55°$   From B: $h = (d+40)\tan35°$

(b)

$d\tan55° = (d+40)\tan35°$; $d(1.42814 - 0.70021) = 40 \times 0.70021$; $d \times 0.72793 = 28.008$; $d \approx \mathbf{38.48 \text{ m}}$

(c)

$h = 38.48 \times \tan55° = 38.48 \times 1.42814 \approx \mathbf{54.95 \text{ m}}$

Interactive

Elevation & Depression — Live Trig Calculator

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Consolidation Game

Angles of Elevation and Depression