How to use the angle converter
- Type a number in the From box. 180 is entered for you, and decimals (22.5) and negative values (-45) both work.
- Pick the unit you have in the first menu and the unit you want in the second. The To box, the result line and the lines under it update as you type.
- To work backwards, type in the To box instead, or press Swap units.
- For degrees, minutes and seconds, use the Decimal degrees box and the Degrees, Minutes and Seconds boxes in the DMS card. Edit either side and the other fills in.
How the conversion works
Every angle is stored as a share of one full turn: 360 degrees, 400 gradians, 21,600 arcminutes, 1,296,000 arcseconds, 6,400 NATO mils, 2π radians and 2,000π milliradians make one turn. The tool computes result = value × To count ÷ From count, which keeps 90° = 100 grad exact. A radian is the angle whose arc is as long as its radius, so 180° = π radians. For DMS, decimal degrees = degrees + minutes ÷ 60 + seconds ÷ 3600, and seconds are rounded to three decimals when splitting. The factors match NIST Guide to the SI, Appendix B.8 and NIST Guide to the SI, Chapter 5. Answers show up to 10 significant digits.
Worked example with the default values
With 180 in the From box, Degree (°) as the From unit and Radian (rad) as the To unit, the result line reads 180° = 3.141592654 rad, because 180 × 2π ÷ 360 = π. Under it you see In terms of π: π rad and the reverse line 1 rad = 57.29577951°. In the DMS card, the default 40.7128° splits as 40° 42′ 46.08″: 0.7128 × 60 = 42.768 minutes, then 0.768 × 60 = 46.08 seconds.
Common angles
| Degrees | Radians | As a decimal | Gradians |
|---|---|---|---|
| 30° | π/6 | 0.5236 | 33.333 |
| 45° | π/4 | 0.7854 | 50 |
| 60° | π/3 | 1.0472 | 66.667 |
| 90° | π/2 | 1.5708 | 100 |
| 180° | π | 3.1416 | 200 |
| 360° | 2π | 6.2832 | 400 |
To get an angle from a slope, use the slope calculator, and for a triangle from two sides and the angle between them, the triangle area calculator. Angular speed in rad/s converts to Hz with the frequency converter, and the unit converter covers other kinds of unit.
Limits and edge cases
Angles are not wrapped to 0 to 360, so 450° stays 450° and equals 1.25 turns. The NATO mil is 1/6400 of a turn, while a true milliradian is 1/6283.19, so they are listed separately. The “In terms of π” line appears only for simple multiples of π with a denominator up to 720. In the DMS card, minutes and seconds should be below 60 and decimal degrees below 1,000,000,000. Letters, currency symbols and slashes are blocked with a hint.
Sources: NIST Guide to the SI, Appendix B.8; NIST Guide to the SI, Chapter 5.
Angle Converter FAQ
How do I convert degrees to radians?
Multiply the degrees by π ÷ 180 (about 0.0174533). 180° is π radians, which is 3.141592654, and 90° is π/2, which is 1.570796327.
How many degrees is one radian?
One radian is 180 ÷ π, or 57.29577951°, which is 57° 17′ 44.81″.
What is a gradian?
A gradian (also called a gon or grad) is 1/400 of a turn, so a right angle is 100 grad and 1 grad is 0.9°. It is used in some surveying, mainly in parts of Europe.
How do I convert decimal degrees to degrees, minutes and seconds?
The whole number is the degrees. Multiply the decimal part by 60 for the minutes, then multiply the new decimal part by 60 for the seconds. 40.7128° becomes 40° 42′ 46.08″.
Why does my calculator or code give strange sine and cosine answers?
Most tools expect radians. In JavaScript, Math.sin(90) returns 0.8939966636, not 1, because 90 is read as 90 radians. Use Math.sin(90 * Math.PI / 180) instead, and check that a calculator is in DEG or RAD mode.