What this calculator does
The angle of depression is the angle measured downward from a horizontal line of sight to a point below and away from the observer. It is the everyday example most trigonometry courses use to introduce inverse tangent: a person standing on a cliff or in a tall building looking down at something on the ground.
This calculator takes the height of the observer above the target and the horizontal distance between them, and returns the angle of depression along with the straight-line, or line-of-sight, distance to the target.
The formula
The height and horizontal distance form the two legs of a right triangle, with the line of sight as the hypotenuse. The angle of depression is the inverse tangent of height divided by horizontal distance, the same calculation as an angle of elevation measured the other way, since the two angles are always equal.
| Term | Meaning |
|---|---|
| Angle of depression | The downward angle from horizontal to the line of sight, at the observer. |
| Angle of elevation | The equal and opposite angle, measured upward from horizontal at the target. |
| Line of sight | The straight-line distance between observer and target, the hypotenuse of the triangle. |
The inputs explained
| Field | What to enter |
|---|---|
| Height above the target (m) | The vertical height of the observer above the target. |
| Horizontal distance to the target (m) | The horizontal distance between the observer's position and the target, measured along the ground. |
When to use it
Sighting from a height
A lookout tower, cliff top or tall building looking down at a fixed point on the ground is the standard setup, with height and horizontal distance both measurable or known in advance.
Aviation and navigation
The angle of depression from an aircraft to a ground reference point is used in approach planning and in classic trigonometry problems involving descent angles.
Surveying and site measurement
Where height above a target is known from a survey and horizontal distance is measured on a plan, the angle of depression gives the sightline angle without needing to measure it directly.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How the angle of depression changes with horizontal distance
A fixed 50 m height above the target, across a range of horizontal distances.
| Horizontal distance | Angle of depression | Line-of-sight distance |
|---|---|---|
| 50 m | 45.00° | 70.71 m |
| 75 m | 33.69° | 90.14 m |
| 100 m | 26.57° | 111.80 m |
| 150 m | 18.43° | 158.11 m |
| 200 m | 14.04° | 206.16 m |
| 300 m | 9.46° | 304.14 m |
Questions
Is the angle of depression the same as the angle of elevation?
They are numerically equal, but measured from opposite ends: the angle of depression is at the observer looking down, and the angle of elevation is at the target looking up, at the same line of sight.
What if I only know the line-of-sight distance and one other side?
The angle can still be found using sine or cosine instead of tangent; this calculator is set up for the common case of a known height and horizontal distance.
Does the angle of depression depend on units?
No, as long as height and horizontal distance are entered in the same unit. The angle itself is a ratio and comes out the same whether both are in metres or both are in feet.
Where else does this calculation come up?
Anywhere a downward or upward sightline needs converting into an angle: ramp and roof pitch problems, camera and telescope angles, and the equivalent angle-of-elevation problems in reverse.
For the ratios this calculation is built from, see the Pythagoras theorem calculator.