Quick answer
The angle of elevation is the angle above the horizontal from an observer up to a point: arctan(height difference ÷ horizontal distance). A height difference of 10 at a horizontal distance of 10 gives 45°.
Enter the height difference and the horizontal distance. The calculator returns the angle of elevation.
Related tools: Angle of Depression Calculator: Angle of depression 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 Elevation 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 elevation | 45° |
Example 2
Inputs: Height difference = 5; Horizontal distance = 8.660254037844386.
| Result | Value |
|---|---|
| Angle of elevation | 30° |
The first line is the headline result (angle of elevation). IEEE-754 double precision; shown to at most 6 decimal places.
Worked example
A flagpole top is 10 m above your eye level and you stand 10 m away horizontally: tan θ = 10 ÷ 10 = 1, so θ = 45°. Standing 20 m away gives arctan(0.5) = 26.57°.
Measuring height from an angle
Reverse the formula to find height: height = distance × tan θ. From 30 m away with an elevation of 35°, the height above eye level is 30 × 0.7002 = 21.0 m; add your eye height for the total.
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 elevation |
|---|---|
| 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 Elevation 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 Depression Calculator](/math/angle-of-depression-calculator/) | height difference and horizontal distance | Angle of depression |
| [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 from the horizontal?
Yes, from the level line through the observer's eye, upward.
How does it relate to the angle of depression?
The angle of depression from the top equals the angle of elevation from the bottom (alternate angles).
What if the distance is along the ground slope?
Use the horizontal distance, not the slope length.
What does the Angle of Elevation Calculator calculate?
Angle of elevation from a height difference and horizontal distance. It returns Angle of elevation.
What formula does the Angle of Elevation 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 Elevation 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 Elevation result?
With height difference 10 and horizontal distance 10, the angle of elevation is 45°.
Related calculators
- Angle of Depression Calculator — Angle of depression 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.
- Tangent Calculator — tan of an angle in degrees or radians.
- Secant Calculator — sec of an angle in degrees or radians.
- Cosecant Calculator — csc 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.