What is Angle Converter (Degrees, Radians, Gradians, Turns)?
degrees to radians, radians to degrees This tool runs entirely in your browser. No data is sent to any server, and no signup is needed.
Degrees to Radians Converter
Convert between degrees, radians, gradians, and turns. Essential for trigonometry, physics, and engineering.
Quick reference: degrees to radians
| degrees | radians |
|---|---|
| 1 degrees | 0.0174533 radians |
| 10 degrees | 0.174533 radians |
| 100 degrees | 1.74533 radians |
| 1,000 degrees | 17.4533 radians |
| 10,000 degrees | 174.533 radians |
About this Angle unit conversion
Convert between degrees, radians, gradians, and turns. Essential for trigonometry, physics, and engineering.
This calculator runs entirely in your browser. Your inputs are never sent to a server. Privacy-first, no tracking, no signup needed.
Frequently asked questions
Why use radians instead of degrees?
Radians are the natural unit for calculus and physics. Derivatives of sin(x) and cos(x) only equal cos(x) and -sin(x) if x is in radians.
How many radians in 360 degrees?
2π radians (about 6.2832). 180° = π radians ≈ 3.1416. 90° = π/2 ≈ 1.5708.
What is a gradian?
A gradian (or gon) divides a right angle into 100 parts. 360° = 400 gradians. Used in some European surveying and military applications.
What is a turn?
One full revolution. 1 turn = 360° = 2π radians = 400 gradians. Common in navigation and gear ratios.
How do I convert degrees to radians manually?
Multiply by π/180 (about 0.01745). E.g. 45° × π/180 ≈ 0.7854 rad.
About angle units
An angle is the rotation between two rays sharing a vertex, measured in degrees, radians, gradians, or turns. The four units describe the same geometric quantity at different scales: a quarter turn is 90 degrees, π/2 radians, 100 gradians, or 0.25 turn. Radians are the SI derived unit for plane angle (BIPM SI Brochure, 9th edition).
Conversion factors
All four units relate through a full revolution. One turn equals 360° equals 2π radians equals 400 gradians (NIST SP 811, Appendix B).
1 degree = π / 180 radian ≈ 0.01745329252 rad 1 radian = 180 / π degree ≈ 57.29577951 degree 1 gradian = 0.9 degree = π / 200 radian 1 turn = 360 degree = 2π radian = 400 gradian
Worked example: 45 degrees to radians
- Start with 45 degrees.
- Multiply by the exact factor π / 180.
- 45 × π / 180 = π / 4.
- π / 4 ≈ 0.7853981634 radian.
Common angles in all four units
| Degree | Radian (exact) | Radian (decimal) | Gradian | Turn |
|---|---|---|---|---|
| 0° | 0 | 0.0000 | 0 | 0 |
| 30° | π/6 | 0.5236 | 33.333 | 0.0833 |
| 45° | π/4 | 0.7854 | 50 | 0.125 |
| 60° | π/3 | 1.0472 | 66.667 | 0.1667 |
| 90° | π/2 | 1.5708 | 100 | 0.25 |
| 180° | π | 3.1416 | 200 | 0.5 |
| 360° | 2π | 6.2832 | 400 | 1 |
Pitfalls and accuracy notes
- Calculator mode: a scientific calculator returns sin(30) = -0.988 in radian mode but 0.5 in degree mode. Always set the mode that matches your input unit.
- Arcseconds and arcminutes: astronomy and surveying subdivide degrees into 60 arcminutes per degree and 60 arcseconds per arcminute. 1 arcsecond = 1/3600 degree ≈ 4.848 microradian.
- NATO milliradian: rounded for fire-control simplicity. NATO defines 1 mil = 1/6400 turn (not 1/6283.185 turn). Russian and Swedish artillery use 1/6000 and 1/6300 respectively.
- π precision: the converter uses π to 12 significant digits. For interferometry, use more digits; for surveying, four are enough.
- Radian is "unitless": formally, radian is a ratio of two lengths so it carries no dimension. The "rad" symbol is optional but recommended for clarity (BIPM SI Brochure).
When to use this converter
- Trigonometry homework: swap between unit-circle radians and textbook degrees.
- Programming: JavaScript's
Math.sintakes radians, but UI inputs are degrees. Convert before calling. - CAD and 3D modelling: Blender and Unity expose both units; check which the API expects.
- Surveying: European surveys often use gradians; convert before sharing with US-style instruments.
Related tools
- Solar Angle Calculator for solar elevation and azimuth.
- Hertz to RPM Converter for rotational frequency.
- Miles to Kilometres for distance conversion.
- Horsepower to Kilowatts for power conversion.
Frequently asked questions
Why is the radian the SI unit and not the degree?
The radian is dimensionless: it is the arc length divided by the radius. That makes derivatives of trig functions clean (d/dx sin(x) = cos(x) only when x is in radians) and avoids the conversion factor in series expansions. The degree is a convention from Babylonian base-60 arithmetic.
How many radians are in a full circle?
2π radians, approximately 6.2831853. Half a circle is π ≈ 3.1416 radians; a right angle is π/2 ≈ 1.5708 radians.
What is a gradian and where is it used?
A gradian (or "gon") divides a right angle into 100 parts, so a full circle is 400 gradians. It is still used in French and some German cadastral surveying because each gradian corresponds to roughly 100 km along a meridian (the original definition tied 10,000 km to 100 gon = π/2 radian, since 10,000 km was the historical metre definition).
How do I convert milliradians to degrees?
1 milliradian = 1/1000 radian = 0.05729578 degree ≈ 3.4377 arcminute. Snipers and artillery use NATO mils where 1 mil = 1/6400 turn, slightly different from the mathematical milliradian.
Does the tool round to 10 decimal places like a calculator?
The widget shows six significant digits to keep the output readable; internal math uses double-precision (15 to 17 significant digits). For exact symbolic answers (like π/3) use a CAS.
Sources
- BIPM (2019), The International System of Units (SI), 9th edition, Table 4 (coherent derived units), definition of radian.
- NIST Special Publication 811 (2008), Guide for the Use of the International System of Units, Appendix B (conversion factors).
- ISO 80000-3:2019, Quantities and units - Part 3: Space and time.
Last updated 2026-05-28.