Key takeaway: The angle of elevation is the angle between the horizontal and the line of sight to an object above, while the angle of depression is measured downward to an object below.
Grade 10 – GeometryG-SRT.C.8Trigonometry
Definition of Angle of Elevation and Depression
The angle of elevation is how far you tilt your gaze upward from level to see something higher, such as a rooftop. Tilting down from level, as when you look from a balcony at a car in the street, gives the angle of depression. Both start at a level line, never at a plumb line.
If two people look at each other, one upward and one downward, their angles match. The level lines through them are parallel, so the two angles are alternate interior angles. This sets up right triangles for heights and distances nobody can reach with a tape.
Formula and notation
With a horizontal distance \(d\), a vertical difference \(h\) and angle \(\theta\), the right-triangle link is:
\[ \tan\theta = \frac{h}{d} \]
Looking down from the upper point gives the same measure as looking up from the lower point.
Key facts
- Both are measured from a level line, not from a sloped surface or the vertical.
- A downward look from above matches the upward look from below.
- Tangent connects the angle to the height and horizontal distance.
- Eye height of the observer must be added when finding a full height.
Angle of Elevation and Depression example
A drone hovers 85 ft above a field, and its camera points \(31^{\circ}\) below level to a parked car. How far is the car from the spot on the ground directly beneath the drone? Use \(\tan 31^{\circ} \approx 0.6009\).
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From the car the angle of elevation to the drone is also \(31^{\circ}\), so in the right triangle the 85 ft height is opposite it.
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Set up: \(\tan 31^{\circ} = \frac{85}{d}\).
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Solve: \(d = \frac{85}{0.6009} \approx 141.5\) ft.
The car is about 141.5 ft from the point beneath the drone.
Common mistake
Watch out
Wes measures the \(31^{\circ}\) from the plumb line hanging below the drone and builds a triangle with 85 ft as the adjacent side.
Why it is wrong
The tilt starts at the level line through the drone, not at the vertical one. Measured from the vertical, the same line of sight would be \(59^{\circ}\). Draw the level line first, then the line of sight below it.
Related terms and lessons
Browse the glossary: A–Z list · Geometry vocabulary · Trigonometry