Angle of Depression Calculator
Enter any two values from the right triangle formed by a downward line of sight - the vertical drop, horizontal distance, or slant (line-of-sight) distance - and this calculator solves for the rest. You can also enter the angle and one distance to find the others. Switch between metric and imperial, and see the full step-by-step working for every result.
What is the angle of depression?
The angle of depression is the angle formed between a horizontal line from an observer's eye and the line of sight going downward to an object below. It is always measured from the horizontal, never from the vertical. When you stand on a cliff and look down at a boat on the water, the angle your line of sight makes with the imaginary horizontal plane at eye level is the angle of depression. By the alternate interior angles theorem, this equals the angle of elevation seen by someone at the boat looking up at you.
The formulas used in this calculator
The angle of depression, vertical drop (h), horizontal distance (d), and slant distance (s) form a right triangle. The four core relationships are: (1) tan(angle) = h / d, so angle = arctan(h / d); (2) sin(angle) = h / s, so angle = arcsin(h / s); (3) cos(angle) = d / s, so angle = arccos(d / s); (4) Pythagorean theorem: s = sqrt(h^2 + d^2). Any two of the four quantities (angle, h, d, s) are enough to determine the other two. This calculator covers all four solve modes - pick which value you need, enter the other two known values, and the result appears instantly.
Angle of depression vs angle of elevation
The angle of depression (looking down) and the angle of elevation (looking up) are always equal for any given observer-object pair, because they are alternate interior angles formed by a transversal (the line of sight) crossing two parallel horizontal lines. This means surveyors, pilots, and engineers can use the same trigonometric formulas regardless of whether they measure from the top looking down or from the bottom looking up. In practice, the choice depends on which measurement is easier to make: an observer on high ground finds the angle of depression, while one at lower elevation measures the angle of elevation.
Real-world applications
Angles of depression appear in surveying (measuring heights of buildings, cliffs, or terrain), aviation (glide paths typically use a 3-degree angle of depression on final approach), nautical navigation (spotting hazards below the horizon), construction (grading slopes and drainage channels), and physics problems involving projectile motion. A standard instrument landing system glide slope of 3 degrees means that for every 1 nautical mile of horizontal distance, an aircraft descends about 318 feet. The same principle governs wheelchair ramp codes (maximum 1:12 slope, approximately 4.76 degrees) and roof pitches specified in rise-over-run ratios.
Common angle of depression reference values
| Angle | Context / example | Vertical : Horizontal ratio |
|---|---|---|
| 1 - 5 deg | Drainage slopes, gentle ramps | 1 : 11 to 1 : 57 |
| 5 - 10 deg | Wheelchair ramps (max ~5 deg), gentle hills | 1 : 6 to 1 : 11 |
| 15 deg | Moderate roof pitch, hiking trail | 1 : 3.73 |
| 26.6 deg | Typical staircase (2:1 run/rise) | 1 : 2 |
| 30 deg | Standard trigonometry example, ski run (black) | 1 : 1.73 |
| 45 deg | 1:1 slope - equal vertical and horizontal | 1 : 1 |
| 60 deg | Very steep terrain, cliff face approach | 1.73 : 1 |
| 90 deg | Looking straight down | Vertical only |
Typical angles of depression encountered in everyday and technical contexts.
Frequently asked questions
What is the difference between angle of depression and angle of elevation?
They are equal in magnitude but measured in opposite directions. The angle of depression is measured downward from the horizontal at the observer's position, while the angle of elevation is measured upward from the horizontal at the lower point. By the alternate interior angles theorem, whenever the two horizontal reference lines are parallel (which they always are if both are level), these angles must be equal. So a 30-degree angle of depression from a cliff top produces a 30-degree angle of elevation from the base of the cliff.
How do I find the angle of depression without a protractor?
If you can measure the vertical drop (height difference) and the horizontal distance, divide the drop by the distance to get the tangent of the angle, then take the arctangent (inverse tangent) using a calculator or table. For example, a 20 m drop over 50 m of horizontal distance gives arctan(20/50) = arctan(0.4) which is approximately 21.8 degrees. Clinometers (inclinometers) are inexpensive hand instruments that measure angles of depression and elevation directly in the field.
Can the angle of depression be greater than 90 degrees?
No. The angle of depression is bounded between 0 degrees (horizontal line of sight - the object is at the same height) and 90 degrees (the observer looks straight down - the object is directly below). Values outside this range do not correspond to a downward line of sight.
Why does the calculator show the angle of elevation equal to the angle of depression?
This follows from the alternate interior angles theorem in geometry. The line of sight acts as a transversal crossing two parallel horizontal lines - the horizontal at the observer and the horizontal at the object. The angle above the lower horizontal (elevation) and the angle below the upper horizontal (depression) are on opposite sides of the transversal between the two parallels, making them alternate interior angles, which are always equal.
What units can I use with this calculator?
Distances can be entered in metres (metric) or feet (imperial) - both give identical angular results because the trig functions only care about the ratio of the two distances. Angles can be displayed in degrees or radians. To convert between them: radians = degrees x pi/180. A 45-degree angle equals pi/4 radians (about 0.7854 rad).
How is the slant (line-of-sight) distance different from horizontal distance?
The horizontal distance is measured along the ground (or projected flat plane) and ignores altitude change. The slant distance is the actual straight-line distance through the air between the observer and the object. The slant is always longer than or equal to both the horizontal and vertical distances. For a 30-degree angle of depression, the slant distance is exactly twice the vertical drop (s = h / sin 30 deg = h / 0.5 = 2h).