Angles of Elevation and Depression: Practical Trigonometry
An angle of elevation is the angle above the horizontal at which you look up at an object, and an angle of depression is the angle below the horizontal at which you look down; both lead to right triangles that can be solved with trigonometry. In brief: if you stand 50 m from a tower and see its top at an angle of elevation of 40°, its height above eye level is 50 · tan 40° ≈ 41.95 m. The angle of depression from the top to you is also 40°.
These angles let you measure heights and distances that cannot be reached with a tape measure: tall buildings, trees, cliffs, aircraft and ships. The method is always the same: draw a right triangle, mark the angle and the known side, choose the ratio and solve. This guide shows how to draw the figures, how elevation and depression are related, gives worked examples including two-step problems and a table of typical results.
What are the angles of elevation and depression?
Both angles are measured from a horizontal line through the observer’s eye. The angle of elevation goes up to the line of sight to an object above. The angle of depression goes down to the line of sight to an object below.
If an observer on a cliff looks down at a boat, the angle of depression of the boat equals the angle of elevation of the observer as seen from the boat. The reason is that the two horizontal lines are parallel, and the angles are alternate angles on the line of sight. This fact is used constantly: draw both horizontals and mark the equal angles.
How do you solve a practical problem step by step?
- Draw a diagram with the observer, the object, the horizontal line and the vertical line forming a right triangle.
- Mark the angle (at the observer’s eye) and the known side.
- Identify opposite, adjacent and hypotenuse relative to the angle.
- Select the ratio: tangent connects height and horizontal distance; sine connects height and the line-of-sight distance; cosine connects horizontal and line-of-sight distance.
- Solve, and add the observer’s eye height if the question asks for the total height.
- Check that the answer is reasonable.
Worked examples
Example 1: height of a tower
A surveyor stands 80 m from the base of a tower. The angle of elevation of the top is 35°, and the instrument is 1.5 m above the ground. The height above the instrument is 80 · tan 35° = 80 · 0.7002 = 56.0 m. The height of the tower is 56.0 + 1.5 = 57.5 m.
Example 2: a lighthouse and a boat
From the top of a lighthouse 45 m above the sea, the angle of depression of a boat is 20°. The horizontal distance is 45/tan 20° = 45/0.3640 = 123.6 m. The line-of-sight distance is 45/sin 20° = 45/0.3420 = 131.6 m.
Example 3: a kite
A kite string of length 60 m makes an angle of elevation of 52° with the horizontal ground. The kite’s height is 60 · sin 52° = 60 · 0.7880 = 47.3 m, and the ground distance to the point under the kite is 60 · cos 52° = 36.9 m.
Example 4: finding the angle
A flagpole is 12 m high. The angle of elevation of its top is measured from a point 9 m from the base (ignoring eye height). tan θ = 12/9 = 4/3, so θ = tan−1(4/3) ≈ 53.13°, and the line of sight has the length 15 m (a 3-4-5 triangle).
Example 5: a slope
A road rises 7 m over each 100 m of road length. The angle is sin−1(7/100) ≈ 4.01°. The grade (rise over horizontal run) is about 7.02%, not 7%, because 100 m is along the road.
How do you solve problems with two observations?
When the distance to the object is unknown, observe it from two points on a line towards it. Let the angles of elevation be α from the farther point and β from the nearer point (β > α), at a distance d apart. Let h be the height and x the distance from the nearer point. Then h = x tan β and h = (x + d) tan α. Solving gives x = d tan α/(tan β − tan α), and h = x tan β.
Example 6
A hill is observed from two points 100 m apart, in line with its top. The angles of elevation are 25° (far) and 40° (near). Then tan 25° = 0.4663 and tan 40° = 0.8391. The distance x from the near point is 100 · 0.4663/(0.8391 − 0.4663) = 46.63/0.3728 = 125.1 m. The height is h = 125.1 · 0.8391 = 105.0 m. Check from the far point: (125.1 + 100) · 0.4663 = 105.0 ✓.
Example 7: a building seen from a window
From a window 10 m above the ground, the angle of elevation of the top of the building opposite is 30°, and the angle of depression of its base is 50°. Let x be the horizontal distance between the buildings. The base is 10 m below the window, so x tan 50° = 10 and x = 10/1.1918 = 8.39 m. The part of the opposite building above the window level is 8.39 · tan 30° = 8.39 · 0.5774 = 4.84 m. So the opposite building is 10 + 4.84 = 14.84 m high.
Height from distance and angle
| Angle of elevation | tan | Height at 100 m | Line-of-sight distance |
|---|---|---|---|
| 10° | 0.1763 | 17.6 m | 101.5 m |
| 20° | 0.3640 | 36.4 m | 106.4 m |
| 30° | 0.5774 | 57.7 m | 115.5 m |
| 40° | 0.8391 | 83.9 m | 130.5 m |
| 45° | 1.0000 | 100.0 m | 141.4 m |
| 50° | 1.1918 | 119.2 m | 155.6 m |
| 60° | 1.7321 | 173.2 m | 200.0 m |
Where does the method come from?
Measuring heights from shadows with similar triangles is ancient: Thales is said to have measured the pyramids in this way. Instruments for angles, such as the astrolabe and the quadrant, were developed by Greek and Islamic astronomers. The great trigonometric survey of India in the nineteenth century used triangulation and precise angle measurements to determine the height of Mount Everest, and the principle is the same as in the examples above.
Where is it used today?
Surveyors, architects and foresters measure heights with a theodolite or a clinometer. Pilots use the angle of depression (glide slope) for landing, typically about 3° for airliners. In sport, the angle of elevation of a launched ball determines its range. Smartphone apps estimate the height of buildings from the camera angle and the distance.
Which mistakes are common?
Measuring the angle from the vertical
The angles of elevation and depression are measured from the horizontal.
Forgetting the observer’s height
The calculation gives the height above eye level. Add the eye height for the total height.
Confusing the angle of depression with the angle at the object
The angle of depression is at the observer, measured down from the horizontal. Inside the triangle, it equals the angle at the object, by alternate angles.
Wrong calculator mode
Make sure that the calculator is set to degrees, not radians.