Elevation/depression | Year 10

01 / LEARN

The idea, step by step

Use the horizontal reference

An elevation angle looks upwards from a horizontal line; a depression angle looks downwards. Neither normally starts at a vertical wall. With parallel horizontal lines, depression and elevation are equal alternate interior angles.

For level-ground height problems, tangent links the vertical difference and horizontal distance. Draw a horizontal sight line from the observer.

Separate the height difference

An instrument 1.5 m above the ground has its horizontal sight line at that height. The triangle gives the extra height above the line; add 1.5 m to find the whole object.

Use degree mode and carry the full calculator result to the final rounding step.

A common trap

Using the full tower height opposite the angle despite a stated eye height.

Instead: Use the height above the horizontal sight line, then add the observer height.

02 / WORKED EXAMPLES

See the thinking

Example 1

Include eye height

An observer stands 20 m from a tower. Eye height is 1.5 m and elevation is 35°. Find tower height to 0.1 m.

$tan 35^{\circ} = (h - 1.5) / 20$ .
$h = 20 tan 35^{\circ} + 1.5$ .

15.5 m

Example 2

Find a distance

A platform is 18 m high. An observer with eye height 1.6 m sees its top at elevation 32°. Find horizontal distance to 0.1 m.

Vertical difference $= 18 - 1.6 = 16.4$ m.
$d = 16.4 / tan 32^{\circ}$ .

26.2 m

Check your understanding

A ground-level sight line has elevation 45° and horizontal distance 12 m. Find the height.

Show answer

12 m

$h = 12 tan 45^{\circ} = 12$ m.

03 / WORKSHEET

Your turn

6 questions · 15 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

From level ground 30 m away, roof elevation is 40°. Find its height to 0.1 m.
2 marks
A tree top is 25 m above an observer sight line, at elevation 50°. Find horizontal distance to 0.1 m.
2 marks
An observer is 16 m from a 12 m wall. Their eye is 1.6 m high. Find elevation to the top to 0.1°.
3 marks
A cliff top is 40 m above sea level. Depression to a boat is 25°. Find the horizontal distance to 0.1 m.
3 marks
A line of sight makes 38° with a vertical wall. Find its angle with the horizontal.
2 marks
A camera 1.2 m high is 15 m from a flagpole. Top elevation is 55°. Find total pole height to 0.1 m.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

From level ground 30 m away, roof elevation is 40°. Find its height to 0.1 m.
2 marks
$h = 30 tan 40^{\circ}$ .
Round the result, not the tangent value.
25.2 m
A tree top is 25 m above an observer sight line, at elevation 50°. Find horizontal distance to 0.1 m.
2 marks
$tan 50^{\circ} = 25 / d$ .
$d = 25 / tan 50^{\circ}$ .
21.0 m
An observer is 16 m from a 12 m wall. Their eye is 1.6 m high. Find elevation to the top to 0.1°.
3 marks
Vertical difference $= 12 - 1.6 = 10.4$ m.
$θ = {tan}^{- 1} (10.4 / 16)$ .
33.0°
A cliff top is 40 m above sea level. Depression to a boat is 25°. Find the horizontal distance to 0.1 m.
3 marks
Use the equal elevation angle at the boat.
$d = 40 / tan 25^{\circ}$ .
85.8 m
A line of sight makes 38° with a vertical wall. Find its angle with the horizontal.
2 marks
$θ = 90^{\circ} - 38^{\circ} = 52^{\circ}$ .
52°
A camera 1.2 m high is 15 m from a flagpole. Top elevation is 55°. Find total pole height to 0.1 m.
3 marks
Height above camera $= 15 tan 55^{\circ}$ .
Add 1.2 m.
22.6 m

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Curriculum reference & lesson scope

Australian Curriculum v9 and South Australian school course scope

Right-triangle applications include AC9M10M03. The approved chapter also develops non-right-triangle methods, exact values and trigonometric functions within one normal Year 10 course.

Official curriculum source ↗

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