Angle of Depression Calculator
Enter the vertical height of the observer above the object and the horizontal distance between them to find the angle of depression — the angle below the horizontal line of sight — plus the slant distance and slope gradient.
m
m
Angle below the horizontal from the observer to the object
- 1
Slope ratio (rise ÷ run)
h ÷ d = 50 ÷ 100 = 0.5 - 2
Slant distance
√(50² + 100²) = 111.8 - 3
Angle of depression
arctan(50 ÷ 100) = 26.57arctan of the slope ratio, measured downward from the horizontal.
How does this calculator work?
Angle of depression θ = arctan(h / d), where h is the vertical drop from observer to object and d is the horizontal distance. It equals the angle of elevation viewed from below. Slant distance = √(h² + d²). A 50 m height and 100 m horizontal distance gives θ ≈ 26.6°.
Formula
How this is calculated
The angle of depression is the angle formed between a horizontal line through the observer's eye and the line of sight down to an object below. If the observer stands at height h above the object's level, with horizontal distance d separating them, the angle is θ = arctan(h / d). It is always measured from the horizontal, not the vertical.
The angle of depression equals the angle of elevation viewed from the object back up to the observer. This is because horizontal lines at different heights are parallel and the line of sight is a transversal — the two angles are alternate interior angles and therefore equal. This symmetry is frequently used in surveying and navigation: you can measure from either end and get the same angle.
The slant distance (the straight-line distance between observer and object) is the hypotenuse: L = √(h² + d²). The slope gradient h/d equals tan(θ) and represents rise per unit of horizontal run — multiply by 100 to get the gradient as a percentage, which is the standard used for road gradients and wheelchair ramp specifications.
Frequently asked questions
Both describe the same line of sight. The angle of depression is measured downward from the horizontal at the higher point; the angle of elevation is measured upward from the horizontal at the lower point. They are always equal because the horizontal lines at the two levels are parallel (alternate interior angles formed by the transversal line of sight).
Measure the height of the viewpoint above the object (h) and the horizontal distance to the object (d). The angle is arctan(h / d). For example, a 50 m high viewpoint with a 100 m horizontal distance gives arctan(50/100) = arctan(0.5) ≈ 26.6°.
Yes. Surveyors use depression angles measured from known heights to infer distances. If you know the depression angle from a theodolite and the height of the instrument above the target, the horizontal distance is d = h / tan(θ). This calculator solves the direct problem — height and distance known, angle unknown.
Also known as
TG we-Calculate Editorial Team. (2026). Angle of Depression Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/angle-of-depression-calculator
TG we-Calculate Editorial Team. "Angle of Depression Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/angle-of-depression-calculator.
TG we-Calculate Editorial Team, "Angle of Depression Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/angle-of-depression-calculator
@misc{wecalculate_angle_of_depression_calculator, title = {Angle of Depression Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/angle-of-depression-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?