Angle of Depression Calculator

How to Calculate Angle of Depression

The angle of depression is the angle between the horizontal line of sight and the line of sight to an object below the horizontal line. It's commonly used in trigonometry and real-world applications such as surveying and navigation.

Formula

The formula to calculate the angle of depression is:

\[ \theta = \tan^{-1}\left(\frac{\text{Vertical Drop}}{\text{Horizontal Distance}}\right) \]

Where:

• θ = Angle of depression
• tan⁻¹ = Inverse tangent (arctangent) function
• Vertical Drop = The height difference between the observer and the object
• Horizontal Distance = The horizontal distance between the observer and the object

Calculation Steps

1. Measure or determine the vertical drop (rise) from the observer to the object.
2. Measure or determine the horizontal distance (run) from the observer to the object.
3. Divide the vertical drop by the horizontal distance.
4. Calculate the inverse tangent (arctangent) of this ratio.
5. Convert the result from radians to degrees if necessary.

Example

Let's calculate the angle of depression for an observer looking at an object with a vertical drop of 30 meters and a horizontal distance of 40 meters.

1. Given:
   • Vertical Drop = 30 meters
   • Horizontal Distance = 40 meters
2. Apply the formula: \[ \theta = \tan^{-1}\left(\frac{30}{40}\right) \]
3. Calculate: \[ \theta = \tan^{-1}(0.75) \] \[ \theta \approx 36.87° \]

Therefore, the angle of depression is approximately 36.87°.

Visual Representation

This diagram illustrates an angle of depression of 36.87° with a vertical drop of 30 meters and a horizontal distance of 40 meters.