Angles of Elevation
Measure how steeply you look UP to see a high point from a lower position. The horizontal is the adjacent; the height is the opposite; the line of sight is the hypotenuse.
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You stand on flat ground and look up at the top of a tree. Sketch the situation, then identify: which side is horizontal, which is vertical, which is your line of sight, and where is the angle of elevation?
The angle of elevation is the angle between the horizontal and your line of sight when you look UP at an object. It's measured FROM the horizontal, never from the vertical.
In the right triangle formed: the horizontal ground is the adjacent; the vertical height (above eye level) is the opposite; the slant line of sight is the hypotenuse. The angle of elevation sits AT YOUR EYE between the horizontal and the line of sight.
Know
- Angle of elevation is measured from horizontal to line of sight, looking UP
- Horizontal = adj, vertical height = opp, line of sight = hyp
- Eye height may need to be added to the final answer
Understand
- The geometry behind elevation problems
- Why the angle is measured from horizontal, not vertical
- How surveyors use this for heights of cliffs, towers, trees
Can Do
- Sketch an elevation problem from a description
- Identify which trig ratio applies
- Solve for either the angle or a side
Wrong: Measuring the angle from the vertical instead of the horizontal — gives 90° minus the right answer.
Right: Angle of elevation is from HORIZONTAL upwards to the line of sight.
Wrong: Forgetting eye height — reporting the calculated opp as the full object height.
Right: The calculated opp is above eye level — add eye height to get total height above ground.
Four-step setup: sketch the situation, mark the angle of elevation, label the three sides, then choose the ratio.
Step 1: Sketch a right triangle with horizontal ground (adj), vertical height (opp), and line of sight (hyp). Step 2: Mark the angle of elevation at the observer's eye. Step 3: Label each side with what's given. Step 4: Use sin/cos/tan based on the side pair.
The opposite side in your triangle is measured from the observer's eye, not the ground. If the question asks for total height ABOVE GROUND, add the eye height.
| Question asks for | Add eye height? |
|---|---|
| Height of object | Yes (if eye not at ground level) |
| Height ABOVE eye level | No |
| Find the angle | No (angle unchanged) |
Watch Me Solve It · 3 examples
- 1Sketch and labeladj = 80, $\theta = 35°$, opp = ?
- 2Use tan$\tan 35° = $ opp$/80$, opp $= 80\tan 35°$
- 3Computeopp $\approx 80 \times 0.7002 \approx 56.01$ m
- 1Set upopp = 25 (height), adj = 40 (ground distance)
- 2Use tan$\tan\theta = 25/40 = 0.625$
- 3Inverse$\theta = \tan^{-1}(0.625) \approx 32.0°$
- 1Height above eyeopp $= 12\tan 50° \approx 12 \times 1.1918 \approx 14.30$ m
- 2Add eye heighttotal $= 14.30 + 1.6 = 15.90$ mThe line of sight starts from eye level.
- 3State answerTree $\approx 15.90$ m tall
Common Pitfalls
Angle of elevation
- From horizontal, looking UP
- Vertex at observer's eye
- Between 0° and 90°
Side labels
- Horizontal = adj
- Vertical (above eye) = opp
- Line of sight = hyp
Eye height
- Add to opp for total height
- Read the question carefully
- State assumption
Common ratio
- Tan most common
- Opp + adj from elevation problems
- sin/cos if hyp involved
How are you completing this lesson?
Brain Trainer · 4 problems
Four quick drills to lock in today's skill. Try each, then reveal the answer.
-
1 40 m away, angle 30°. Height?
opp = $40\tan 30° \approx 23.09$.$\approx 23.09$ m -
2 Height 20 m, distance 50 m. Angle?
$\tan^{-1}(20/50) \approx 21.8°$.$\approx 21.8°$ -
3 100 m away, angle 25°. Height?
$100\tan 25° \approx 46.63$.$\approx 46.63$ m -
4 Line of sight 50 m at 30°. Height (opp)?
opp = $50\sin 30° = 25$.25 m
Quick Check · 5 questions
Show Your Working · 3 questions
Q6. Standing 50 m from the base of a building, the angle of elevation to the top is 38°. (a) Find the height of the building above eye level. (b) If eye height is 1.6 m, find the building's total height (2 d.p.).
Q7. A surveyor 70 m from a tower measures the angle of elevation to the top as 28°. Find the tower's height above eye level.
Q8. From a point on flat ground, a tower is seen at angle of elevation 40°. Walking 30 m further away (in the same straight line), the angle drops to 25°. Find the height of the tower (2 d.p.). [Hint: let $d$ = original distance; set up two equations and solve.]
Quick Check
1. A — From horizontal upward.
2. B — $60\tan 45° = 60$.
3. C — $\tan^{-1}(2/3) \approx 33.7°$.
4. D — Adjacent.
5. A — 57.74 + 1.5 = 59.24.
Show Your Working Model Answers
Q6 (3 marks): (a) opp $= 50\tan 38° \approx 39.07$ m [2]. (b) Total $= 39.07 + 1.6 = 40.67$ m [1].
Q7 (2 marks): opp $= 70\tan 28°$ [1] $\approx 37.23$ m [1].
Q8 (4 marks): Let $d$ = original distance. $h = d\tan 40°$ and $h = (d+30)\tan 25°$ [1]. Equate: $d\tan 40° = (d+30)\tan 25°$ [1]. $d(\tan 40° - \tan 25°) = 30\tan 25°$. $d = 30\tan 25°/(\tan 40° - \tan 25°) \approx 13.99/0.372 \approx 37.62$ m [1]. $h = 37.62 \tan 40° \approx 31.57$ m [1].
Two angles, one tower
From point A, a building is seen at 40° elevation. From point B, 50 m further from the building along the same horizontal line, the angle is 25°. Find the building's height (2 d.p.).
Reveal solution
Let $d_A$ = distance from A and $h$ = height. $h = d_A \tan 40°$. From B: $h = (d_A + 50)\tan 25°$. Equate: $d_A(\tan 40° - \tan 25°) = 50\tan 25°$. $d_A \approx 23.32/0.372 \approx 62.69$ m. $h \approx 62.69 \tan 40° \approx 52.61$ m.
Definition
From horizontal, looking UP
Horizontal = adj
Always the adjacent
Vertical = opp
Above eye level
Line of sight = hyp
Slants upward
Tan is common
opp and adj as legs
Eye height adds
If asked for above-ground height
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