Angle of Depression Calculator
Calculate the angle of depression from horizontal distance and vertical drop. Includes interactive diagram, reverse solving, and trigonometric formulas.
About
The angle of depression is the angle formed between a horizontal line of sight and the line of sight downward to a target object. It equals arctan(h ÷ d), where h is the vertical drop and d is the horizontal distance. Miscalculating this angle in surveying, navigation, or structural engineering leads to misaligned foundations, incorrect slope grades, or failed line-of-sight installations. This calculator solves for any unknown variable given two knowns and renders a proportional geometric diagram. It assumes a flat Earth approximation valid for distances under 50 km and does not account for atmospheric refraction.
Applications range from determining the pitch angle for drainage pipes (minimum 1° slope per most building codes) to computing the look-down angle for surveillance cameras or the glide slope for aircraft approach paths (typically 3°). Note: the angle of depression from point A to point B equals the angle of elevation from point B to point A, by alternate interior angles with a horizontal transversal. Pro tip: always verify whether your measurement references the horizontal or the vertical before plugging values into any formula.
Formulas
The angle of depression θ is computed from the vertical drop (opposite side) and horizontal distance (adjacent side) using the inverse tangent function:
Where θ = angle of depression in degrees, h = vertical drop (height difference) in any consistent length unit, and d = horizontal distance in the same unit.
The line-of-sight distance (hypotenuse) is derived from the Pythagorean theorem:
Radian-to-degree conversion uses:
Reverse solving uses standard trigonometric identities. Given θ and d: h = d × tan(θ). Given θ and h: d = htan(θ). The slope grade percentage is computed as: G = hd × 100%.
Reference Data
| Application | Typical Angle | Standard / Reference | Notes |
|---|---|---|---|
| Aircraft Glide Slope (ILS) | 3° | ICAO Annex 10 | Standard precision approach |
| Drainage Pipe (Minimum) | 0.6° | IPC / UPC Plumbing Codes | 1⁄8 inch per foot |
| Drainage Pipe (Recommended) | 1.2° | IPC / UPC Plumbing Codes | 1⁄4 inch per foot |
| Wheelchair Ramp (ADA) | 4.76° | ADA Standards 405.2 | Max 1:12 slope ratio |
| Staircase (Residential) | 30 - 35° | IRC R311.7 | Rise 7.75in, run 10in |
| Road Grade (Highway Max) | 3.4° | AASHTO Green Book | 6% grade on freeways |
| Road Grade (Mountain) | 5.7 - 8.5° | AASHTO | 10 - 15% grade |
| Roof Pitch (Low Slope) | 9.5° | IRC R905 | 2:12 pitch |
| Roof Pitch (Standard) | 18.4 - 26.6° | IRC R905 | 4:12 to 6:12 |
| Ski Slope (Green / Beginner) | 6 - 14° | NSAA Classification | 10 - 25% grade |
| Ski Slope (Blue / Intermediate) | 14 - 22° | NSAA Classification | 25 - 40% grade |
| Ski Slope (Black / Expert) | 22 - 40° | NSAA Classification | >40% grade |
| Security Camera Look-Down | 15 - 45° | ASIS Guidelines | Optimal facial recognition range |
| Surveying Theodolite Range | 0 - 90° | ISO 17123-3 | Vertical angle measurement |
| Solar Panel Tilt (Equator) | 10 - 15° | IEC 61215 | From horizontal surface |
| Solar Panel Tilt (45°N Lat) | 30 - 45° | IEC 61215 | Seasonal adjustment required |
| Artillery Firing (Depression) | 0 - 5° | FM 6-40 | Downhill engagement correction |
| Cliff Angle (Geological) | 45 - 90° | USGS Classification | ≥45° classified as cliff |
| Escalator (Standard) | 30° | EN 115-1 | Max 35° for rise ≤ 6m |
| Conveyor Belt (Max) | 18° | CEMA 7th Ed. | Bulk material without cleats |