What is the Angle of Depression?
The angle of depression is the angle between the horizontal line and the line of sight when an observer looks downward toward an object. In other words, it is how much a line deviates from the horizontal in the downward direction.
This concept is widely used in construction, surveying, navigation, and everyday trigonometry problems — for example, calculating the slope of a ramp, the inclination of a roof, or the viewing angle from a cliff to a boat.
Angle of Depression Formula
The angle of depression is found using the arctangent (arctan) function:
θ = arctan(vertical distance / horizontal distance)
Where:
- θ — angle of depression (in degrees)
- vertical distance — the height difference between the observer and the object
- horizontal distance — the horizontal distance from the observer to the point directly above or below the object
The formula works because the observer, the object, and the reference horizontal form a right triangle. The vertical and horizontal distances are the two legs of this triangle, and arctan gives us the angle opposite the vertical leg.
How to Use This Calculator
- Choose your unit system: Metric (m, km, cm, mm) or US Imperial (ft, in, yd, mi).
- Select the specific distance unit for both measurements.
- Enter the horizontal distance — the flat ground distance between the two points.
- Enter the vertical distance — the height (drop) between the observer and the object.
- Click Calculate.
The calculator returns the angle of depression in degrees, in radians, and as a slope percentage.
Sample Calculation
Suppose you are standing on a cliff and looking down at a boat. The horizontal distance from you to the boat is 200 m, and the cliff is 80 m high.
θ = arctan(80 / 200) = arctan(0.4) ≈ 21.8°
So the angle of depression from you to the boat is approximately 21.8°.
Angle of Elevation vs. Angle of Depression
The angle of elevation and the angle of depression are always equal when measured between the same two points — they are alternate interior angles formed by a transversal crossing two parallel horizontal lines. So if the angle of depression from the cliff to the boat is 21.8°, the angle of elevation from the boat to the cliff top is also 21.8°.
Unit Systems Supported
Metric
- m — meters (most common for construction and engineering)
- km — kilometers (for large-scale surveys)
- cm — centimeters
- mm — millimeters
US / Imperial
- ft — feet (standard for US construction)
- in — inches
- yd — yards
- mi — miles
Since the angle depends on the ratio of vertical to horizontal distance, both values must be in the same unit.
FAQs
Can the angle of depression be greater than 90°?
No. The angle of depression is defined between 0° and 90°. A value of 0° means a perfectly horizontal line of sight, while 90° means you are looking straight down.
What if the horizontal and vertical distances are equal?
If both distances are equal, the ratio is 1, and arctan(1) = 45°.
Is slope percentage the same as the angle?
No. The slope percentage is tan(θ) × 100%. For example, a 45° angle corresponds to a 100% slope, while a 30° angle corresponds to about a 57.7% slope.
What tools measure the angle of depression directly?
Surveying instruments such as a clinometer, theodolite, or total station can measure angles of depression directly in the field.