Angle of Depression Triangle Calculator and Formula

An angle of depression triangle calculator finds the downward angle from a horizontal line of sight to an object below it. Enter the horizontal distance and the vertical drop, and it returns the angle in degrees, along with the line-of-sight distance.

Formula

The calculator models a right triangle. The horizontal distance is the adjacent side to the angle, and the vertical drop is the opposite side. Therefore:

Angle of depression, θ = arctan(vertical drop ÷ horizontal distance)
In symbols: θ = tan⁻¹(v / h)

• θ is the angle of depression, in degrees.
• v is the vertical difference in height between the observer and the target, in metres in this calculator.
• h is the horizontal distance between the observer and the target, also in metres.

The line-of-sight distance, or hypotenuse, is s = √(h² + v²). The calculator reports this as a useful companion measurement. If you enter both distances in feet instead, the angle is unchanged and the line-of-sight result is in feet: the two distances simply need to use the same unit.

Worked Example

Using the calculator’s default values, the horizontal distance is 40 m and the vertical drop is 30 m.

1. Divide the vertical drop by the horizontal distance: 30 ÷ 40 = 0.75.
2. Take the inverse tangent: θ = arctan(0.75) ≈ 36.87°.
3. Find the line-of-sight distance: s = √(40² + 30²) = √(1,600 + 900) = √2,500 = 50.00 m.

So an observer 30 m above a point 40 m away sees it at an angle of depression of 36.87°, and the straight-line distance to it is 50 m.

What an Angle of Depression Means

An angle of depression is measured downward from a horizontal line through the observer. It is not measured from the vertical wall, the ground, or the sloping line of sight. Imagine looking straight ahead, then lowering your gaze toward an object: the angle between those two directions is the angle of depression.

In a standard diagram, the observer is at the upper end of a right triangle, the target is below, and the horizontal leg runs from the observer’s level toward the target. The line of sight forms the hypotenuse. The angle of depression is at the observer, between the horizontal and the hypotenuse.

A related angle often causes confusion: the angle of elevation from the target to the observer is equal to the angle of depression when the two horizontal reference lines are parallel. They are alternate interior angles formed by the line of sight crossing those parallel lines. The angles have different viewpoints, but the same numerical measure.

When to Use the Calculator

Use this calculation when a right-triangle setup gives a horizontal separation and a vertical height difference, and you need the downward viewing angle. It is useful for textbook geometry and for estimating sight lines in a simplified, level-ground model. For example, if a lookout point is 18 ft above a point on the ground and that point is 24 ft horizontally away, the angle is arctan(18/24), about 36.87°.

The calculator needs the vertical drop, not necessarily the observer’s height above the ground. If the target is also elevated, subtract its height from the observer’s height to get the vertical difference. For example, an observer at 20 m and a target at 5 m have a 15 m vertical drop, assuming both heights are measured from the same reference level.

How to Measure and Read the Inputs

• Horizontal distance: Measure the level, horizontal separation between the points, not the distance along a slope or the length of the sight line.
• Vertical drop: Use the difference in elevation from the observer to the target. A target at the same height has a drop of zero.
• Use matching units: The example calculator uses metres for both inputs. Convert feet to metres if mixing measurements, or calculate with both in feet outside the calculator. Since the ratio v/h is unitless, matching units produce the same angle.
• Read the angle in degrees: The output is an angle, while the second result is a distance. Do not treat the line-of-sight distance as a horizontal distance.

The result is a geometric angle, not a compass direction or a complete three-dimensional bearing. It describes the vertical plane containing the observer and target.

Common Errors and Limits

A frequent mistake is using the inverse sine or cosine instead of inverse tangent. With the horizontal and vertical legs known, tangent is the direct relationship: tan(θ) = opposite ÷ adjacent = v/h. Sine or cosine can also work if you first know the hypotenuse, but that is not required here.

Do not swap the numerator and denominator. The angle is arctan(vertical drop ÷ horizontal distance), not arctan(horizontal distance ÷ vertical drop). Swapping them gives the complementary angle in a right triangle. Also check that your calculator or software displays the answer in degrees, not radians.

The result depends on accurate measurements and the assumption that the horizontal and vertical directions form a right angle. If the ground slopes, a measured distance along it is not automatically the horizontal leg. Likewise, real-world surveying over long distances can require corrections for terrain, instrument alignment, and Earth curvature; this simple triangle does not account for those effects. A zero horizontal distance makes the formula undefined, and a zero vertical drop gives an angle of 0°.

Frequently Asked Questions

How do you calculate an angle of depression?
Divide the vertical drop by the horizontal distance, then take the inverse tangent: θ = arctan(v/h). Make sure the calculator is set to degrees if you want the answer in degrees.

Is angle of depression the same as angle of elevation?
They are measured from different locations: depression from the observer looking down, elevation from the lower point looking up. In the usual right-triangle diagram, they are equal because they are formed by a transversal crossing parallel horizontal lines.

Can I use feet instead of metres?
Yes. Use feet for both the horizontal distance and vertical drop. Their ratio gives the same angle as metre measurements, while the calculated line-of-sight distance will be in feet.

What if I know the line-of-sight distance instead?
If you know the hypotenuse and the vertical drop, use θ = arcsin(v/s). If you know the hypotenuse and horizontal distance, use θ = arccos(h/s). These are equivalent right-triangle relationships, provided the measurements describe the same triangle.

Can the angle of depression be greater than 90°?
For an object below the observer, the usual angle of depression is between 0° and 90°. A result outside that range usually means the points or the angle convention have been entered or interpreted incorrectly.

Related

• Calculate m²: Square a Number or Measure Area
• Calculate Triangle Height: Formulas and Examples
• Calculate Equation: Formula, Steps, and Examples

in h | Horizontal distance | m | 40 | 0
in v | Vertical drop | m | 30 | 0
out angle | Angle of depression | degrees | atand(v / h) | 2
out sight | Line-of-sight distance | m | sqrt(h^2 + v^2) | 2