Angle of Depression Calculator
Use this angle of depression calculator to find the downward viewing angle from a horizontal line of sight, or solve for the horizontal or vertical distance in the same right triangle.
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Result
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Quick Answer: The angle of depression is found with angle = arctan(vertical distance / horizontal distance). The same relation can be rearranged to find either distance when the angle is known.
How to Calculate
- Set up the right triangle: Identify the vertical drop and horizontal run that describe the line of sight.
- Apply the tangent relation: Use the ratio vertical divided by horizontal to find the tangent of the angle.
- Take the inverse tangent: Use arctangent to convert the ratio into the angle of depression.
- Rearrange when needed: If the angle is known, use tangent to solve for the missing vertical or horizontal distance.
Formula
angle = arctan(vertical / horizontal)
| Variable | Meaning | Unit |
|---|---|---|
| angle | Angle of depression | degrees |
| vertical | Vertical drop | any consistent length unit |
| horizontal | Horizontal distance | same as vertical |
Worked Examples
Angle - Lookout height problem
- Vertical drop: 30 m
- Horizontal distance: 50 m
Result: Angle of depression = 30.96°
The line of sight drops by just under 31 degrees below the horizontal.
Horizontal distance - Find run from angle
- Vertical drop: 20 m
- Angle: 25°
Result: Horizontal distance = 42.89 m
A smaller angle creates a longer horizontal run for the same drop.
Vertical distance - Find drop from angle
- Horizontal distance: 100 ft
- Angle: 15°
Result: Vertical drop = 26.79 ft
The drop remains relatively small because the viewing angle is shallow.
Interpretation Table
| Range | Meaning | Action |
|---|---|---|
| Small angle | Shallow downward sight line | A long horizontal distance often drives shallow depression angles. |
| Moderate angle | Balanced run and drop | Typical of many textbook trigonometry and site-geometry problems. |
| Large angle | Steep downward sight line | The vertical drop is large relative to the horizontal distance. |
Frequently Asked Questions
Yes, in a standard right-triangle setup they are equal as alternate interior angles when the horizontal lines are parallel.
Use tangent. If you know the angle and the vertical drop, horizontal distance = vertical / tan(angle). If you know the angle and horizontal distance, vertical drop = horizontal x tan(angle).
It is the angle measured downward from a horizontal line of sight to the object being observed.
It is useful in surveying, site layout, lookout problems, and any right-triangle setup involving a downward line of sight.
Note: This calculator assumes a flat horizontal reference and a simple right-triangle geometry.
References
Last reviewed: March 14, 2026