Angle of Elevation Calculator

How to Calculate Angle of Elevation and Depression

The angle of elevation is the angle formed between a horizontal line and a line of sight to an object above the horizontal line. Conversely, the angle of depression is formed when looking at an object below the horizontal line. These concepts are fundamental in trigonometry and have practical applications in fields such as surveying, navigation, and architecture.

Formula

The formula to calculate both the angle of elevation and depression is:

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

Where:

• θ = Angle of elevation or depression
• tan⁻¹ = Inverse tangent (arctangent) function
• Vertical Height = 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 height (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 height 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 for an observer looking at an object with a vertical height of 15 meters and a horizontal distance of 20 meters.

1. Given:
   • Vertical Height = 15 meters
   • Horizontal Distance = 20 meters
2. Apply the formula: \[ \theta = \tan^{-1}\left(\frac{15}{20}\right) \]
3. Calculate: \[ \theta = \tan^{-1}(0.75) \] \[ \theta \approx 36.87° \]

Therefore, the angle is approximately 36.87°. If the observer is looking up at the object, this is the angle of elevation. If looking down, it's the angle of depression.

Visual Representation

This diagram illustrates an angle of 36.87° with a vertical height of 15 meters and a horizontal distance of 20 meters.