Looking up and looking down
- The angle of elevation is the angle from the horizontal up to your line of sight.
- The angle of depression is the angle from the horizontal down to your line of sight.
Both are always measured from the horizontal, never from the vertical.
They’re equal
If a person on a cliff looks down at a boat, and a person on the boat looks up at the cliff top:
angle of depression from the cliff = angle of elevation from the boat
They are alternate angles between two parallel horizontal lines. This fact is very useful.
Measuring angles
A clinometer (a protractor with a weighted string, or a phone app) measures these angles in the field. Surveyors use more precise instruments called theodolites.
Example: elevation
From 30 m away, the angle of elevation to the top of a tree is 40°. How tall is the tree?
- tan 40° = h/30
- h = 30 tan 40° ≈ 25.2 m
If the observer’s eye is 1.6 m above the ground, add it: ≈ 26.8 m.
Example: depression
A lighthouse keeper 45 m above the sea sees a boat at an angle of depression of 12°. How far is the boat from the base of the lighthouse?
- The angle at the boat (elevation) is also 12°.
- tan 12° = 45/d
- d = 45/tan 12° ≈ 211.7 m
Example: two angles
From the top of a 60 m building, the angles of depression to two cars in a straight line are 30° and 45°. How far apart are the cars?
- Closer car: 60/tan 45° = 60 m
- Farther car: 60/tan 30° ≈ 103.9 m
- Distance apart ≈ 43.9 m
Example: finding an angle
A 20 m tower casts a 35 m shadow. What is the angle of elevation of the sun?
- tan θ = 20/35
- θ ≈ 29.7°
Drawing the diagram
- Draw the horizontal line from the observer.
- Draw the line of sight.
- Mark the angle between them.
- Complete the right triangle with a vertical line.
- Label known and unknown lengths.
Worked examples
Example 1: A plane at 2,000 m altitude is seen at a 15° angle of elevation. Ground distance? 2,000/tan 15° ≈ 7,464 m.
Example 2: From a 10 m window, the angle of depression to a dog is 25°. Distance to the dog along the ground? 10/tan 25° ≈ 21.4 m.
Example 3: A kite is 50 m high on a 70 m string. Angle of elevation? sin⁻¹(50/70) ≈ 45.6°.
Common mistakes
- Measuring from the vertical instead of the horizontal.
- Putting the angle of depression inside the triangle at the top, where it doesn’t belong (use alternate angles).
- Forgetting eye height when it’s given.
Try it yourself
- Angle of elevation 60°, distance 15 m. Height?
- From a 100 m cliff, angle of depression to a ship is 8°. Distance?
- A 1.7 m person sees the top of a 12 m pole at an angle of elevation. They stand 18 m away. Angle?
Answers: 1) ≈ 26.0 m 2) ≈ 711.5 m 3) tan⁻¹(10.3/18) ≈ 29.8°
Quick recap
- Angle of elevation: up from the horizontal; angle of depression: down from it.
- The two are equal (alternate angles).
- Draw a right triangle and use tan, sin or cos.