Arctan Calculator
Find the angle whose tangent is a given value, in degrees and radians. Results between −90° and 90°.
arctan(1) = 45°
arctan(1)
45°
In radians
0.7854 rad
Multiply degrees by π/180
Arctan is the inverse of tangent: it returns the angle whose tangent is the given value. The result always lies between −90° and 90°.
Last updated: August 2026
How this calculator is verified
Checked by True Calculator automated test suite on
- Formula verified against a published worked example in the automated test suite
- Edge cases (zero, negative, boundary and unit-mismatch inputs) covered by unit tests
The full verification method is on our how we verify page. Found an error? Tell us and we will re-check it.
When to Use This Calculator
Arctan turns any rise-and-run ratio into the angle people actually picture, which is why it dominates Indian civil-engineering and surveying work. Road builders convert gradient percentages to degrees, ramp designers check that a wheelchair ramp stays under 5°, and railway engineers use it to compute track superelevation angles. Farmers and irrigation teams measure canal slopes by comparing fall over length, and contractors set drain pipes with a calculated angle instead of guesswork. Drone pilots compute camera tilt from altitude and ground distance, photographers dial in the same angle from object height, and astronomy enthusiasts find the elevation of celestial objects from shadow ratios. Students use arctan to solve 'find the angle of elevation' questions in every board exam, where the ratio is given and the angle is the answer.
How to Use This Calculator
- Step 1: Enter any tangent value — positive, negative or zero, such as 1.
- Step 2: Read the angle in degrees whose tangent equals that value.
- Step 3: Read the same angle in radians alongside it.
- Step 4: Check that the result lies between −90° and 90°, which is always true for arctan.
Worked Example
A road contractor in Himachal Pradesh builds a short stretch that rises 3 m over a horizontal run of 12 m. The gradient ratio is 3 ÷ 12 = 0.25, and arctan(0.25) returns about 14.04°, the slope angle drivers will climb. A perfectly balanced ratio of 1, as when a 5 m pole casts a 5 m shadow, gives arctan(1) = 45°, the classic one-to-one slope. The calculator also confirms arctan(0) = 0° for flat ground and arctan(−1) = −45° for downward slopes, while very large inputs approach but never reach 90°.
Tips and Common Mistakes
- •Tip 1: Divide rise by run before entering — arctan takes the ratio, not the two measurements separately.
- •Tip 2: A negative tangent means a downward slope, and the answer is a negative angle.
- •Tip 3: For road gradients, the tangent value equals the percentage rise when the run is 100 — a 3% grade means arctan(0.03) ≈ 1.72°.
- ✗Mistake 1: Entering the run divided by the rise instead of rise divided by run, which gives the wrong angle.
- ✗Mistake 2: Forgetting that arctan always answers between −90° and 90°, so a 100° slope cannot exist in its output.
Frequently Asked Questions
What is arctan?
Arctan is the inverse of tangent: it answers 'which angle has this tangent value?' Since tan(45°) = 1, arctan(1) = 45°. It is also written as tan⁻¹.
Does arctan accept any number?
Yes — because tangent can take any real value, arctan accepts any finite input. arctan(0) = 0°, arctan(√3) ≈ 60°, and very large inputs approach 90° without ever reaching it.
What is the range of arctan?
The result always lies between −90° and 90° (−π/2 to π/2 radians). arctan(−1) = −45°, and the extremes approach ±90° as the input grows.
When would I use arctan in real life?
For slope angles: a road that rises 3 m over a horizontal run of 12 m has a gradient angle of arctan(3/12) ≈ 14.04°. It is also used for camera field-of-view and direction finding from coordinates.
How do I convert an arctan result to a slope?
The tangent of the angle gives the rise per unit run. An arctan result of 45° corresponds to a slope of 1, meaning 1 unit up for every 1 unit across — the steepest angle you can build stairs at comfortably.
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