Input data in angle of depression calculator to find the angle formed between the horizontal line of sight and the line of sight to an object located below the observer.
For example, a lifeguard in a tower spots a swimmer in distress. The angle of depression helps determine the swimmer’s distance from the shore, enabling a more efficient rescue operation.
Angle of Depression Calculator
| Height (m) | Distance (m) | Angle (degrees) | Angle (radians) |
|---|---|---|---|
| 10 | 20 | 26.57° | 0.4636 rad |
| 50 | 30 | 59.04° | 1.0303 rad |
| 100 | 150 | 33.69° | 0.5880 rad |
| 75 | 80 | 43.15° | 0.7532 rad |
| 200 | 100 | 63.43° | 1.1071 rad |
Conversion equation: Radians = Degrees × (π / 180)
Angle of Depression Formula
The formula is :
tan(θ) = Height / Distance
Where:
- θ is the angle of depression
- Height is the vertical distance between the observer and the object
- Distance is the horizontal distance from the base of the observer’s position to the object
If a drone operator is flying a drone at 100 meters altitude and spots a target 200 meters away horizontally, the angle of depression would be:
tan(θ) = 100 / 200 = 0.5
θ = arctan(0.5) ≈ 26.57°
The formula for calculating the angle of depression is derived from trigonometric principles. In a right-angled triangle formed by the observer’s position, the object, and the ground, the angle of depression (θ) is given by:
How to Find Distance in Angle of Depression?
To find the distance using the angle of depression, we rearrange the formula:
Distance = Height / tan(θ)
This calculation is particularly useful in scenarios where direct measurement is impractical or impossible.
Consider a situation where a hiker on a mountain ridge wants to estimate the distance to a lake below. If the hiker knows their elevation above the lake is 500 meters and measures an angle of depression of 30°, the distance to the lake would be:
Distance = 500 / tan(30°) ≈ 866.03 meters
Can an Angle of Depression be 90 Degrees?
Theoretically, an angle of depression can approach 90 degrees, but it can never exactly reach it in practical scenarios.
A 90-degree angle of depression would imply that the observer is directly above the object, with no horizontal separation.
In reality, even when looking straight down from a tall structure or aircraft, there’s always a slight deviation from 90 degrees due to factors like:
- The observer’s eye position relative to the edge of the structure
- The physical impossibility of positioning oneself exactly above a point on the ground
For example, a skydiver in free fall might experience an angle of depression very close to 90 degrees, but it would still be slightly less due to the horizontal movement caused by wind and the Earth’s rotation.
How to Find the Angle of Depression in a Right Triangle
To find the angle of depression in a right triangle:
- Identify the known sides: Usually, you’ll know the height (opposite side) and the horizontal distance (adjacent side).
- Use the tangent function: tan(θ) = Opposite / Adjacent
- Calculate the inverse tangent (arctan) to find the angle
For example, if you’re standing on a 30-meter tall building and spot a car 40 meters away from the base:
- Opposite (height) = 30 meters
- Adjacent (distance) = 40 meters
- tan(θ) = 30 / 40 = 0.75
- θ = arctan(0.75) ≈ 36.87°
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