Look up and look down. Elevation and depression connect the ground to the sky through right-angled triangles. Master the art of drawing diagrams from words.
Today's hook: You stand 50 metres from the base of the Sydney Tower Eye and look up at the top. You cannot measure the height directly, but with the angle of elevation and your distance from the base, you can calculate it without leaving the ground.
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You stand 50 metres from the base of the Sydney Tower Eye. You look up at the top. What information would you need to calculate how tall the tower is? Sketch a quick diagram.
Record your answer in your workbook.
1
The Big Idea
+5 XP to read
Angle of elevation looks up from the horizontal. Angle of depression looks down from the horizontal. Both are measured from a horizontal line through the observer's eye never from the ground or a wall.
The crucial fact: elevation from A to B equals depression from B to A. They are alternate angles between parallel horizontal lines. This lets you swap a depression problem for an equivalent elevation problem, whichever is easier to work with.
Elevation from A up to B equals depression from B down to A: they are alternate angles between parallel horizontals.Text description
Two parallel horizontal lines, each marked with a parallel arrow and labelled horizontal: a lower one through point A and an upper one through point B.
A straight line of sight joins A (lower left) to B (upper right).
At A, the angle of elevation is marked between the lower horizontal and the line of sight, above the horizontal.
At B, the angle of depression is marked between the upper horizontal and the line of sight, below the horizontal.
The two angles are alternate angles between parallel lines, so they are equal.
Elevation and depression are measured from a horizontal line, never from the ground.
They are equal
Elevation from A to B = depression from B to A. Alternate angles rule.
Draw first
Never calculate before sketching. The diagram reveals the ratio.
2
What You'll Master
objectives
Know
The definitions of angle of elevation and angle of depression
That elevation from A to B equals depression from B to A
How to label opposite, adjacent and hypotenuse in elevation/depression diagrams
Understand
How to translate a word description into a right-angled triangle diagram
Why elevation and depression are alternate angles between parallel horizontals
Can Do
Draw a diagram for elevation and depression problems
Calculate heights and distances using trig ratios
Solve multi-step problems involving both elevation and depression
3
Words You Need
vocabulary
Angle of elevationThe angle measured upward from the horizontal to the line of sight.
Angle of depressionThe angle measured downward from the horizontal to the line of sight.
Line of sightThe straight line from the observer's eye to the object being viewed.
Alternate anglesAngles on opposite sides of a transversal cutting parallel lines; elevation and depression are alternate angles.
HorizontalPerfectly level, parallel to the ground. All elevation and depression angles are measured from this line.
VerticalPerpendicular to the horizontal. Forms the right angle in elevation and depression triangles.
4
Spot the Trap
heads-up
Wrong: Drawing the angle of elevation from the ground up to the top of the object, not from the horizontal.
Right: Always measure elevation and depression from the horizontal line through the observer's eye.
Wrong: Using the angle of depression directly in a triangle at the bottom instead of recognising it equals the angle of elevation.
Right: Angle of depression from top = angle of elevation from bottom. They are alternate angles between parallel horizontals.
5
Parts of the Whole
+5 XP
Every elevation and depression problem contains the same six pieces: an observer, an object, a horizontal line through the observer's eye, a line of sight, a vertical drop from the object, and the angle between the horizontal and the line of sight.
Before you choose a ratio, identify the observer and object, draw the horizontal and line of sight, then drop the vertical to form the right angle. The triangle is now exposed.
Draw the horizontal through the observer first; θ is measured down from it to the line of sight. The vertical from the object makes the right angle.Text description
The observer is a point above the ground, with a dashed horizontal line drawn through it.
A line of sight runs from the observer down to the object on the ground.
The angle θ is marked between the dashed horizontal and the line of sight, below the horizontal.
A vertical line runs from the object straight up to the dashed horizontal, and a small square marks the right angle where it meets the horizontal.
The horizontal, the vertical and the line of sight form the right-angled triangle. θ is inside it at the observer: the vertical (height) is opposite θ and the horizontal (distance) is adjacent, so tan θ = height ÷ distance.
Mark where the person is standing or looking from.
Mark the right angle
The vertical drop from the object meets the horizontal at 90°.
Name the unknown
Call the height $h$ or the distance $d$ before calculating.
6
Draw the Diagram
+5 XP
The most important skill in this lesson is translating words into a diagram. Follow the same four steps every time and you will never lose track of which side is which.
Step 1: Draw the horizontal line through the observer's eye. Step 2: Draw the line of sight to the object. Step 3: Mark the angle between horizontal and line of sight. Step 4: Draw the vertical from the object down to the horizontal.
Draw in this order every time; only when the triangle is complete do you label opposite, adjacent and hypotenuse.Text description
Step 1, Horizontal: draw a dashed horizontal line through the observer.
Step 2, Line of sight: draw the line from the observer to the object.
Step 3, Mark angle: mark θ between the horizontal and the line of sight.
Step 4, Vertical drop: draw the vertical from the object to the horizontal, forming the right angle where they meet.
The sketch shows an observer above the ground with each of these four parts, marked with badges 1 to 4, and a small square at the right angle.
Strip: then label O, A, H.
elevation = depression
Draw horizontal first
Everything is measured from this line. Get it right first.
Line of sight is key
It connects observer to object and carries the angle.
Vertical makes 90°
The vertical drop from the object creates the right angle.
7
Choose the Ratio
+5 XP
Once the diagram is drawn, the ratio choice is automatic. Identify which two sides you know and which side you want, then match to SOH CAH TOA.
In most elevation and depression problems, you know the horizontal distance and the angle, and you want the vertical height. That is opposite over adjacent tangent. If you know the hypotenuse (like a string or ladder length), use sine or cosine instead.
Height and distance are the opposite and adjacent sides of θ, so the ratio is tan θ = h ÷ d.Text description
A right-angled triangle with the right angle, marked by a small square, at the bottom left.
The angle θ is marked at the bottom-right vertex, on the ground.
The vertical side is labelled opposite (height): it is opposite θ.
The side along the ground is labelled adjacent (distance): it runs from θ to the right angle.
The sloping side is labelled hypotenuse: it is the line of sight.
Strip: O + A = tan, so tan θ = h ÷ d.
$\tan \theta = \frac{h}{d}$
Find the right triangle
Every elevation problem hides one right-angled triangle.
Label O, A, H
Mark the sides inside your diagram before choosing.
Most problems use tan
Height and distance are opposite and adjacent, tan.
8
Check and Verify
+5 XP
After solving, run a three-point check: the units match, the answer is reasonable, and the elevation equals the depression if you look at the problem from the other end.
A quick sanity check catches most errors. If you are 40 m from a tree and the elevation is 32°, the height should be less than 40 m (tan 32° < 1). A height of 250 m would be impossible. If you used depression instead of elevation, check that the alternate angle gives the same answer.
$h = d \times \tan \theta$
Catch unit mix-ups
Distance in km and height in m will give wrong answers.
Small angle = small height
If tan θ < 1, the height is less than the distance.
Swap observer and object
The alternate angle test confirms your setup is correct.
Watch Me Solve It · 3 examples
Watch Me Solve It · Angle of elevation
+15 XP per step
Q1
PROBLEM
A person stands 40 m from the base of a tree. The angle of elevation to the top is $32°$. Find the height of the tree to the nearest metre.
Check: tan 25° is about 0.5, so d should be about 60 m. Close enough.
Nice work, XP earned
Answer$d \approx 64$ m
Watch Me Solve It · Firefighter ladder
+15 XP per step
Q3
PROBLEM
A firefighter's ladder is 12 m long and leans against a building at an angle of elevation of $68°$. (a) How high up the building does the ladder reach? (b) How far from the base of the building is the foot of the ladder? (c) Is it safe if regulations require the angle to be between $65°$ and $75°$?
Regulations are about the angle, not the height or distance.
Nice work, XP earned
Answer(a) $h \approx 11.1$ m (b) $d \approx 4.5$ m (c) Safe
9
Common Pitfalls
heads-up
Measuring from the ground instead of horizontal
Drawing the angle from the ground up to the object. Elevation is always measured from a horizontal line through the observer's eye.
Fix: draw the horizontal line first, then measure the angle between that line and the line of sight.
Forgetting alternate angles
Trying to use the angle of depression directly inside a triangle at ground level. Depression lives at the top; you need to transfer it to the bottom using alternate angles.
Fix: write "depression = elevation" on your diagram, then work with the elevation angle at the base.
Using the wrong trig ratio
Knowing the hypotenuse (ladder length) but using tan anyway. If you have hypotenuse, you need sin or cos.
Fix: label O, A, H on your diagram before choosing. Hypotenuse = sin or cos. No hypotenuse = tan.
Copy Into Your Books
Definitions
Elevation: angle up from horizontal
Depression: angle down from horizontal
Both measured from observer's eye level
Key Fact
Elevation from A to B = Depression from B to A
They are alternate angles
Parallel horizontal lines, one transversal
Drawing Steps
1. Horizontal through observer
2. Line of sight to object
3. Mark the angle
4. Vertical drop for right angle
Ratio Choice
Height + distance = tan
Hypotenuse known = sin or cos
Label O, A, H first
How are you completing this lesson?
Brain Trainer · 4 problems
D
Brain Trainer · Mixed
4 problems
Four elevation and depression problems. Work each one, then reveal the answer.
1 A person stands 30 m from a building. The angle of elevation to the top is $40°$. Find the height.
Correct, elevation and depression are alternate angles between parallel horizontals.
Not quite, think about parallel horizontal lines and a transversal.
Explanation: The horizontal line through A and the horizontal line through B are parallel. The line of sight AB is a transversal. Angle of elevation (at A, above horizontal) and angle of depression (at B, below horizontal) are alternate angles, so they are equal.
MC2
Find the height
+10 XP
A person stands 30 m from a building. The angle of elevation to the top is $40°$. The height is closest to:
Correct, $h = 30 \times \tan 40° \approx 25$ m.
Not quite, use $\tan 40° = \frac{h}{30}$ and solve for $h$.
Explanation: $\tan 40° = \frac{h}{30}$, so $h = 30 \times \tan 40° \approx 30 \times 0.839 \approx 25.2$ m. The closest answer is 25 m.
MC3
Depression to distance
+10 XP
From a window 20 m high, the angle of depression to a car is $30°$. The horizontal distance is:
Correct, depression = 30° elevation, so $d = \frac{20}{\tan 30°} \approx 35$ m.
Remember, depression from top equals elevation from bottom.
Explanation: The angle of depression from the window equals the angle of elevation from the car (alternate angles). So $\tan 30° = \frac{20}{d}$ and $d = \frac{20}{\tan 30°} \approx \frac{20}{0.577} \approx 34.6$ m. The closest answer is 35 m.
MC4
Ramp angle
+10 XP
A ramp rises 2 m over a horizontal distance of 15 m. The angle of elevation is closest to:
Explanation: $\tan \theta = \frac{2}{15} = 0.133$, so $\theta = \tan^{-1}(0.133) \approx 7.6°$. The closest answer is 8°.
MC5
Plane and airport
+10 XP
A plane is flying at 3000 m. The angle of depression to an airport is $5°$. The horizontal distance is closest to:
Correct, $d = \frac{3000}{\tan 5°} \approx 34,300$ m.
A small angle of depression means a very large horizontal distance.
Explanation: $\tan 5° = \frac{3000}{d}$, so $d = \frac{3000}{\tan 5°} \approx \frac{3000}{0.0875} \approx 34,286$ m. That's about 34.3 km. A small depression angle from a great height means the object is far away horizontally.
Short Answer · 3 questions
Q6
Firefighter ladder safety
+15 XP
Q6
SHORT ANSWER
A firefighter's ladder is 12 m long and leans against a building at an angle of elevation of $68°$. (a) How high up the building does the ladder reach? (b) How far from the base of the building is the foot of the ladder? (c) Is it safe if regulations require the angle to be between $65°$ and $75°$?
(c) $68°$ is between $65°$ and $75°$, so it is safe.
Q7
Surveyor and mountain
+15 XP
Q7
SHORT ANSWER
A surveyor stands at point A and measures the angle of elevation to the top of a mountain as $15°$. After walking 500 m towards the mountain to point B, the angle of elevation is $25°$. (a) Draw a diagram showing both triangles. (b) By letting the height of the mountain be $h$ and the distance from B to the mountain be $x$, write two equations using tangent. (c) Solve for $h$ to the nearest metre.
Write your working in your book.
(a) Diagram with two right-angled triangles sharing height $h$. From A, the horizontal distance is $x + 500$. From B, the horizontal distance is $x$.
(b) $\tan 25° = \frac{h}{x}$ and $\tan 15° = \frac{h}{x+500}$.
A drone is flying at a height of 80 m. The pilot at ground level sees the drone at an angle of elevation of $35°$. (a) Calculate the horizontal distance from the pilot to the point directly below the drone. (b) The drone flies horizontally away from the pilot, maintaining the same height. When the angle of elevation is $20°$, how far has the drone travelled horizontally from its original position? (c) Explain why the angle of elevation decreases as the drone moves away, even though its height stays the same.
(b) New distance: $\tan 20° = \frac{80}{d_2}$ → $d_2 = \frac{80}{\tan 20°} \approx \frac{80}{0.364} \approx 220$ m.
Distance travelled = $220 - 114 = \mathbf{106}$ m.
(c) As the drone moves away, the adjacent side increases while the opposite side (height) stays constant. Since $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$, a larger denominator with the same numerator gives a smaller value, so the angle decreases.
S
Stretch Challenge · Two-triangle mountain
+20 XP
S
STRETCH
A surveyor stands at point A and measures the angle of elevation to the top of a mountain as $15°$. After walking 500 m towards the mountain to point B, the angle of elevation is $25°$. Find the height of the mountain to the nearest metre. Show all working including the two equations and their solution.
Record in your book, full marks require clear working.
Let $h$ = height of mountain, $x$ = distance from B to mountain base.
From B: $\tan 25° = \frac{h}{x}$ → $h = x \tan 25°$.
Use the interactive below to explore elevation and depression scenarios. Switch between modes, adjust the angle, and see how the height and distance change. Try to predict the missing value before revealing it.
Record two observations about how elevation and depression relate to each other.
Lesson Complete
+10 XP
Mark this lesson as complete to earn your bonus XP.