Angle of Depression Calculator

Calculate the angle of depression, elevation, or slope between two points. Perfect for surveying, construction, navigation, and trigonometry problems. Enter any two values to find the third.

Vertical height of the observer above the object

Horizontal distance from observer to the object

Angle of depression from horizontal line of sight

Angle of Depression: θ = arctan(height / distance) Where: • θ = Angle of depression (degrees) • height = Vertical distance between observer and object • distance = Horizontal ground distance Angle of Elevation = Angle of Depression (Alternate interior angles) Slope Grade: Grade (%) = (height / distance) × 100 Line of Sight (hypotenuse): LOS = √(height² + distance²) Height:Distance Ratios: • 1:1 → 45° (100% grade) • 1:2 → 26.57° (50% grade) • 1:3 → 18.43° (33.3% grade) • 1:4 → 14.04° (25% grade) • 1:10 → 5.71° (10% grade) For distance given angle and height: distance = height / tan(θ) For height given angle and distance: height = distance × tan(θ)
Example 1: Looking Down from a Building Standing on a 100 ft tall building, looking at a car 200 ft away: θ = arctan(100/200) = 26.57° Grade = 50% Line of Sight = √(100² + 200²) = 223.6 ft Ratio = 1:2 Example 2: Finding Distance to a Boat Viewing from a 150 ft cliff Angle of depression to boat: 15° Distance = 150 / tan(15°) = 559.8 ft Line of Sight = √(150² + 559.8²) = 579.5 ft The boat is about 560 ft offshore. Example 3: Finding Building Height Standing 300 ft from a building Angle of elevation to top: 30° Height = 300 × tan(30°) = 173.2 ft Angle of depression from top to you: 30° (Same angle, alternate interior!) Example 4: Road Grade Road climbs 50 ft over 1000 ft horizontal Angle = arctan(50/1000) = 2.86° Grade = 5% This is a typical highway grade (5%). Steep roads may reach 10-15%. Common Slopes: • Airport runway: 1-1.5% • Highway: 3-6% • Residential street: 5-10% • Sidewalk ramp (ADA): 8.33% max • Ski slope beginner: 10-25% • Ski slope expert: 50-80% Angle Equivalents: • 1° ≈ 1.75% grade • 5° ≈ 8.75% grade • 10° ≈ 17.6% grade • 26.6° ≈ 50% grade (1:2) • 45° = 100% grade (1:1)

What is the angle of depression and how is it different from the angle of elevation?

The angle of depression is the angle formed between the horizontal line of sight and the line of sight when looking downward at an object. The angle of elevation is the same concept but looking upward. Crucially, when the line of sight connects two points, the angle of depression from the higher point equals the angle of elevation from the lower point (alternate interior angles in parallel lines). Both are measured from the horizontal.

How do I calculate the angle of depression?

The angle of depression θ = arctan(opposite/adjacent) = arctan(height/distance), where height is the vertical distance between observer and object, and horizontal distance is the ground distance separating them. For example, if you are 100 ft tall and an object is 200 ft away horizontally, θ = arctan(100/200) = arctan(0.5) = 26.57°. This works regardless of the units as long as they are consistent.

What is the relationship between angle of depression and slope?

The tangent of the angle of depression equals the slope (grade) expressed as a decimal: tan(θ) = height/distance. A 45° depression angle corresponds to a 100% slope (1:1 ratio). A 26.57° angle is a 50% slope (1:2 ratio). This relationship is used extensively in surveying, civil engineering, and road design where grades are expressed as percentages.

What practical applications use the angle of depression?

Angle of depression calculations are essential in: (1) Surveying - measuring building heights and land elevations using a theodolite; (2) Aviation - calculating descent angles and approach paths; (3) Navigation - determining distances to landmarks from ships; (4) Architecture - designing sightlines for stadiums and theaters; (5) Photography - calculating the field of view from elevated positions; (6) Military - targeting and rangefinding.

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