Quick answer
The angle of depression is the angle below the horizontal from an observer looking down to a point: arctan(height difference ÷ horizontal distance). A drop of 10 over a horizontal distance of 10 gives 45°.
Enter the height difference and the horizontal distance. The calculator returns the angle of depression.
Related tools: Angle of Elevation Calculator: Angle of elevation from a height difference and horizontal distance. Reference Angle Calculator: Acute angle between the terminal side and the x-axis. Coterminal Angle Calculator: Angles sharing the same terminal side.
How to use the Angle of Depression Calculator
- Enter height difference (above 0).
- Enter horizontal distance (above 0).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Height difference | A number | > 0 |
| Horizontal distance | A number | > 0 |
Formula and method
Formula
angle = arctan(height ÷ distance)
IEEE-754 double precision; shown to at most 6 decimal places.
Right-triangle trigonometry; height and distance use the same unit.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Height difference = 10; Horizontal distance = 10.
| Result | Value |
|---|---|
| Angle of depression | 45° |
Example 2
Inputs: Height difference = 5; Horizontal distance = 8.660254037844386.
| Result | Value |
|---|---|
| Angle of depression | 30° |
The first line is the headline result (angle of depression). IEEE-754 double precision; shown to at most 6 decimal places.
Worked example
From a cliff 10 m above the water, a boat 10 m out horizontally is seen at tan θ = 1, i.e. 45° below the horizontal. A boat 50 m out is at arctan(0.2) = 11.31°.
Equal to the elevation from below
The line of sight from the top forms alternate angles with the line of sight from the boat, so the angle of depression from the top equals the angle of elevation from the boat. This lets you use either calculator for the same triangle.
How the result changes with height difference
Holding every other input at the first worked example, this table changes only height difference. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Height difference | Angle of depression |
|---|---|
| 5 | 26.565051° |
| 10 | 45° |
| 20 | 63.434949° |
| 40 | 75.963757° |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Height difference: entering "0" shows "Height difference must be greater than 0." beside that field and no result.
Choosing between the Angle of Depression Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Angle of Elevation Calculator](/math/angle-of-elevation-calculator/) | height difference and horizontal distance | Angle of elevation |
| [Reference Angle Calculator](/math/reference-angle/) | angle | Reference angle |
| [Coterminal Angle Calculator](/math/coterminal-angle/) | angle | Smallest non-negative coterminal angle |
| [Double Angle Calculator](/math/double-angle-calculator/) | angle and unit | sin 2θ |
| [Half Angle Calculator](/math/half-angle-calculator/) | angle and unit | sin(θ/2) |
Background: trigonometry
In a right triangle, sin θ is the opposite side over the hypotenuse, cos θ the adjacent side over the hypotenuse and tan θ the opposite over the adjacent. Their reciprocals are cosecant, secant and cotangent. At 30° the values are sin = 0.5, cos = 0.866025 and tan = 0.577350.
Check the angle unit before you calculate: degrees convert to radians by multiplying by π/180, so 30° is about 0.5236 rad. Some values are undefined, for example tan 90°, because the cosine is zero there and the ratio would divide by zero.
Assumptions and limitations
- Positive values only
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for height difference to confirm this.
Frequently asked questions
Is it measured upward or downward?
Downward from the horizontal.
How do I find the horizontal distance?
distance = height ÷ tan θ; a 10 m drop at 20° is 27.5 m away.
Can the angle exceed 90°?
No, depression angles are between 0° and 90°.
What does the Angle of Depression Calculator calculate?
Angle of depression from a height difference and horizontal distance. It returns Angle of depression.
What formula does the Angle of Depression Calculator use?
angle = arctan(height ÷ distance). Right-triangle trigonometry; height and distance use the same unit.
What do I need to enter in the angle of Depression Calculator?
You enter Height difference and Horizontal distance.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Positive values only.
What is an example of a angle of Depression result?
With height difference 10 and horizontal distance 10, the angle of depression is 45°.
Related calculators
- Angle of Elevation Calculator — Angle of elevation from a height difference and horizontal distance.
- Reference Angle Calculator — Acute angle between the terminal side and the x-axis.
- Coterminal Angle Calculator — Angles sharing the same terminal side.
- Double Angle Calculator — Double-angle identities evaluated at an angle.
- Half Angle Calculator — Half-angle identities evaluated at an angle.
- Trigonometry Calculator — All six trigonometric functions of an angle.
- Sine Calculator — sin of an angle in degrees or radians.
- Tangent Calculator — tan of an angle in degrees or radians.
- Secant Calculator — sec of an angle in degrees or radians.
Sources
- Standard trigonometric definitions on the unit circle (NIST DLMF §4.14)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.