Angle of depression is arctangent of rise over run
Standing above something and looking down, the angle below your horizontal line of sight is the angle of depression. It depends only on the vertical drop and the horizontal distance, through the tangent.
Height is the vertical difference between you and the object, not your absolute elevation. If you are at 50 ft and the object at 20 ft, the height is 30 ft.
Solve for whichever value you're missing
The same right triangle gives you the other pieces. Rearrange the tangent, and use the Pythagorean length for the line of sight.
- Height from angle and distance: height = distance × tan(angle).
- Horizontal distance from angle and height: distance = height ÷ tan(angle).
- Slope distance (line of sight): √(height² + distance²).
Angle to slope, at a glance
| Angle | Height : distance | As a grade |
|---|---|---|
| 2.9° | 1 : 20 | 5% |
| 5.7° | 1 : 10 | 10% |
| 14° | 1 : 4 | 25% |
| 26.6° | 1 : 2 | 50% |
| 45° | 1 : 1 | 100% |
Road grade is just the tangent as a percent: a 10% grade is arctan(0.10) ≈ 5.7°.
The mistakes that flip the answer
- Sine instead of tangent — with the flat run in hand, it is tan = height / distance. Sine uses the slope length, so it gives a different, wrong angle here.
- Slope for run — if your distance is the line of sight, not the ground distance, convert first (distance = slope × cos(angle)) before using tangent.
Common questions
How do I calculate the angle of depression?
Take the arctangent of the height divided by the horizontal distance: angle = arctan(height / distance). Height is the vertical drop from your eye to the object, and distance is measured flat along the ground, not along the line of sight.
Is the angle of depression the same as the angle of elevation?
Numerically, yes. The angle you look down at an object equals the angle it would look up at you, because they are alternate angles across the same horizontal line. They differ only in direction: depression looks down, elevation looks up.
Should I use tangent, sine, or cosine?
Use tangent when you have height and horizontal distance, since tan(angle) = height / distance. Sine and cosine come in only when your known distance is the slope (line of sight) rather than the flat run.
