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Trigonometry · Trigonometry: Similar Triangles, Right Triangles and Acute Triangles

Angles of Elevation and Depression: Measuring What You Can't Reach

2.3About 75 minutes
Learning goals
  • I can identify angles of elevation and angles of depression in a diagram.
  • I can explain why an angle of depression equals the matching angle of elevation.
  • I can use the primary trigonometric ratios to find heights and distances I cannot measure directly.
  • I can build and use a clinometer to measure an angle of elevation.

Words to know

angle of elevation
The angle between the horizontal and your line of sight when you look up at an object.
angle of depression
The angle between the horizontal and your line of sight when you look down at an object.
line of sight
The straight line from your eye to the object you are looking at.
clinometer
A tool for measuring angles of elevation and depression.
inaccessible
Impossible or unsafe to reach and measure directly.

Looking up and looking down

The curriculum asks you to find "the height of an inaccessible object around the school". Think flagpoles, light standards, the school roof or a tall maple tree. You cannot climb them, but you can stand on the ground, measure a distance, measure one angle, and let trigonometry do the rest.

Every such problem uses one of two angles. Both are measured from the horizontal, never from the vertical.

  • The angle of elevation is the angle up from the horizontal to your line of sight.
  • The angle of depression is the angle down from the horizontal to your line of sight.

If a lookout on a cliff looks down at a boat, and someone in the boat looks up at the lookout, both use the same line of sight. The two horizontal lines are parallel, so the angle of depression and the angle of elevation are alternate angles, and therefore equal.

horizontalangle of depressionangle of elevationline of sight
The angle of depression from the lookout equals the angle of elevation from the boat, because the dashed horizontal lines are parallel (alternate angles).
Worked example

Example 1: Height of a flagpole

Jess stands 30 m from the base of a flagpole on level ground. Her eyes are 1.6 m above the ground. The angle of elevation to the top of the pole is 25°. How tall is the flagpole?

  1. Draw the right triangle at eye level: horizontal leg 30 m (adjacent to 25°), vertical leg x (opposite).
  2. tan 25° = x/30, so x = 30 tan 25° = 13.989... m.
  3. The triangle starts at eye level, so add the eye height: 13.989 + 1.6 = 15.589...
  4. Round: about 15.6 m.

The flagpole is about 15.6 m tall.

25°30 mx1.6 meye
Example 1, drawn to scale. The right triangle sits on Jess's eye level, so the eye height (1.6 m) must be added at the end.
Worked example

Example 2: Angle of depression

A lookout platform is 45 m above a lake. The angle of depression from the platform to a canoe is 12°. How far is the canoe from the base of the platform, horizontally?

  1. The angle of elevation from the canoe to the platform is also 12° (alternate angles). Use the triangle at the water.
  2. The height 45 m is opposite 12°; the distance d is adjacent. tan 12° = 45/d.
  3. d = 45 ÷ tan 12° = 211.7...

The canoe is about 212 m away.

Worked example

Example 3: Two answers from one angle

A drone hovers 80 m above level ground. From the operator's position on the ground, the angle of elevation to the drone is 35°. Find (a) the horizontal distance to the point under the drone and (b) the straight-line distance to the drone. (Ignore the operator's height.)

  1. (a) 80 m is opposite 35°, d is adjacent: d = 80 ÷ tan 35° = 114.25... ≈ 114 m.
  2. (b) The line of sight is the hypotenuse: s = 80 ÷ sin 35° = 139.47... ≈ 139 m.
  3. Check: the hypotenuse (139 m) is longer than both legs. ✓

(a) about 114 m; (b) about 139 m.

Worked example

Example 4: Finding the angle

A communications tower is 50 m tall. You stand on level ground 120 m from its base. What is the angle of elevation to the top (ignore your height)?

  1. tan θ = 50/120.
  2. θ = tan⁻¹(50/120) = 22.6...° ≈ 23°.

About 23°.

Making a clinometer

A clinometer measures angles of elevation. You can make one from a protractor, a drinking straw, tape, string and a small weight (a metal nut or washer).

  1. Tape the straw along the straight edge of the protractor.
  2. Tie the string through the small hole at the centre of the straight edge and tie the weight to the other end, so it hangs past the curved edge.
  3. Look through the straw at the top of an object. The string hangs straight down.
  4. Read where the string crosses the scale. The angle of elevation is the difference between that reading and 90°.
eyesight along the straw →to the top of the objectstring reads 120°angle of elevation= 120° − 90° = 30°weight
A homemade clinometer tilted up 30°. When the straw is level the string reads 90°. Here it reads 120°, so the angle of elevation is 30°.
Just for fun

Why is a clinometer such a good friend?

It always looks up to you, and it never takes your problems at face value: it measures them from every angle.

Hands-on activity

Measure something tall at your school or in your neighbourhood

You need: your homemade clinometer, a measuring tape (or count paces: measure one of your paces first), a partner and a notebook. Choose a safe place away from traffic.

  1. Choose a tall object on level ground: a flagpole, a light standard, a basketball hoop or the school wall.
  2. Stand some distance away. Measure the horizontal distance to the base of the object.
  3. Your partner measures the height of your eyes above the ground.
  4. Sight the top of the object with the clinometer and record the angle of elevation. Take three readings and average them.
  5. Calculate: height = distance × tan(angle) + eye height.
  6. Repeat from a different distance and compare.

What to look for: a labelled sketch, the eye height added at the end, and two answers that agree within about 10%. For a basketball rim (3.05 m is the standard height) you can check your answer exactly.

Your turn

1

Sam stands 25 m from a tree. The angle of elevation to the top is 38° and his eyes are 1.5 m above the ground. How tall is the tree?

2

From the top of a 60 m cliff, the angle of depression to a canoe is 20°. How far is the canoe from the base of the cliff?

3

A building is 40 m tall. From a point on level ground 75 m from its base, what is the angle of elevation to the top?

4

A plane is flying at an altitude of 1500 m. The angle of depression from the plane to the airport is 9°. What is the horizontal distance to the airport, in kilometres?

5

A kite string is 50 m long, held 1.2 m above the ground, and makes an angle of elevation of 55°. How high is the kite?

6

Explain why the angle of depression from A to B equals the angle of elevation from B to A.

More practice: warm-up

1

What is an angle of elevation?

2

What is an angle of depression?

3

From a boat, the angle of elevation to the top of a lighthouse is 15°. What is the angle of depression from the top of the lighthouse to the boat?

4

You know the horizontal distance and the angle of elevation and want the height. Which ratio do you use?

5

Why do you add your eye height in Example 1?

More practice: on your own

1

A ski-hill chairlift cable is 850 m long and rises at an angle of 24°. How much height does the lift gain?

2

From a boat 200 m from the base of a cliff, the angle of elevation to the top of a lighthouse on the cliff is 18°. How high is the top of the lighthouse above the water?

3

Ana's eyes are 1.6 m above the ground. She sees the top of a 12 m flagpole at an angle of elevation of 30°. How far is she from the pole?

4

From an apartment window 24 m above the ground, the angle of depression to a parked car is 32°. How far is the car from the building?

5

A 15 m tree casts a 22 m shadow. What is the angle of elevation of the sun?

Challenge

1

The angle of elevation of the sun is 40°. How long is the shadow of a 1.8 m tall person? At what angle of elevation would the shadow be exactly as long as the person is tall?

2

From the roof of a 30 m building, the angle of depression to the base of a taller building across the street is 35°, and the angle of elevation to its top is 20°. How far apart are the buildings, and how tall is the taller one?

3

In Example 1, suppose Jess forgot to add her eye height. By what percent would her answer be too low?

Check your understanding

Quiz

1.

An angle of elevation is measured:

2.

From a platform 45 m above a lake, the angle of depression to a canoe is 12°. About how far away (horizontally) is the canoe?

3.

Standing 30 m from a pole, eyes 1.6 m high, the angle of elevation to the top is 25°. How tall is the pole?

4.

Why does an angle of depression equal the matching angle of elevation?

5.

A tower 50 m tall is 120 m away on level ground. What is the angle of elevation to its top (ignore eye height)?

6.

In a protractor clinometer, the string reads 90° when:

For parents and teachers

Big idea: Expectation 2.3: solving real-life problems with right-triangle trigonometry, including "the height of an inaccessible object around the school".

How to help: Build the clinometer together (protractor, straw, string, washer) and go measure a flagpole or a basketball net. Stay away from roads, and remind your teen never to look at the sun when sighting. Ask them to draw a sketch with the horizontal line, the angle and the eye height before calculating.

Common difficulties:

  • Measuring the angle from the vertical instead of the horizontal.
  • Forgetting to add the eye height (or adding it when the angle was measured from the ground).
  • Putting the angle of depression inside the wrong triangle. Use the equal angle of elevation at the other end instead.
  • Mixing up the horizontal distance and the line-of-sight distance.