📐 Angle Converter
By ToolNimba Editorial Team · Updated 2026-06-19
Type in any box and every other unit updates instantly.
This angle converter switches a value between degrees, radians, gradians, arcminutes, arcseconds and revolutions. Type a number into any box and every other unit updates at once, so you can move from a degree measurement to the radians a calculator or trig function expects without doing the arithmetic by hand. It is handy for geometry homework, programming, surveying, navigation and astronomy work.
What is the Angle Converter?
An angle measures the amount of turn between two rays that share an endpoint. The same turn can be written in several units, which is why a single rotation reads as 360 degrees, 2 pi radians, 400 gradians or 1 revolution depending on the system. Each unit is just a different sized slice of a full circle, so converting between them is a matter of scaling by a fixed factor.
The two units you meet most often are degrees and radians. Degrees split a full circle into 360 equal parts, a convention inherited from ancient Babylonian astronomy. Radians instead measure an angle by the arc length it sweeps on a unit circle, so a full circle is 2 pi radians and a half circle is pi radians. Radians are the natural unit in calculus and physics because formulas for derivatives and series come out cleanly only when angles are in radians.
The smaller and larger units fill specific niches. Gradians (also called gons) divide a circle into 400 parts and are used in some surveying. Arcminutes and arcseconds split a single degree into 60 and 3600 parts and are the standard for the tiny angles in astronomy, optics and precise navigation. This tool converts every value through degrees as a common pivot, so any pair of units lines up correctly.
When to use it
- Converting a degree value into radians before calling a trig function in JavaScript, Python or a spreadsheet, since most languages expect radians.
- Turning the radians output of a physics or calculus problem back into degrees that are easier to picture.
- Reading a surveying angle given in gradians and expressing it in degrees for a drawing.
- Breaking a small astronomical or optical angle into arcminutes and arcseconds.
How to use the Angle Converter
- Type the angle you have into the box for its unit (for example, enter 90 in the Degrees box).
- Read the equivalent value from every other unit box, which updates as you type.
- Use the 90, 180 or 360 preset buttons to load a common degree value quickly.
- Press Clear to empty all boxes and start a new conversion.
Formula & method
Worked examples
Convert 90 degrees to radians.
- radians = degrees × pi ÷ 180
- radians = 90 × 3.14159265 ÷ 180
- radians = 282.743 ÷ 180
Result: 90° = 1.5707963 rad (pi ÷ 2)
Convert 2 radians to degrees.
- degrees = radians × 180 ÷ pi
- degrees = 2 × 180 ÷ 3.14159265
- degrees = 360 ÷ 3.14159265
Result: 2 rad = 114.591559°
Convert 0.5 degrees to arcminutes and arcseconds.
- arcminutes = degrees × 60 = 0.5 × 60 = 30
- arcseconds = degrees × 3600 = 0.5 × 3600 = 1800
Result: 0.5° = 30 arcminutes = 1800 arcseconds
Common angles across units
| Degrees | Radians | Gradians | Revolutions |
|---|---|---|---|
| 0° | 0 | 0 | 0 |
| 30° | π/6 ≈ 0.5236 | 33.333 | 0.0833 |
| 45° | π/4 ≈ 0.7854 | 50 | 0.125 |
| 90° | π/2 ≈ 1.5708 | 100 | 0.25 |
| 180° | π ≈ 3.1416 | 200 | 0.5 |
| 360° | 2π ≈ 6.2832 | 400 | 1 |
How each unit divides a full circle
| Unit | Parts in a full circle | 1 degree equals |
|---|---|---|
| Degree | 360 | 1 degree |
| Radian | 2π ≈ 6.2832 | 0.0174533 rad |
| Gradian | 400 | 1.11111 gon |
| Arcminute | 21,600 | 60 arcminutes |
| Arcsecond | 1,296,000 | 3600 arcseconds |
| Revolution | 1 | 0.00277778 turn |
Common mistakes to avoid
- Feeding degrees into a function that expects radians. Trig functions in most programming languages and spreadsheets take radians, not degrees. Passing 90 instead of pi ÷ 2 (about 1.5708) gives wildly wrong results. Convert first, or use a degrees mode if one is offered.
- Confusing gradians with degrees. A right angle is 90 degrees but 100 gradians. Because the numbers look similar near small angles, it is easy to mix them up. Check which unit your instrument or output is actually using.
- Mixing up arcminutes and minutes of time. The arcminute symbol (a single prime) also appears for minutes of time and feet. An arcminute is one sixtieth of a degree, not a unit of time. Read the context before converting.
- Rounding pi too early. Using 3.14 instead of a fuller value of pi introduces a visible error in radian conversions. This tool keeps full precision and only rounds the displayed result, so do the same in your own calculations.
Glossary
- Degree
- An angle unit equal to one 360th of a full circle. Written with the degree symbol.
- Radian
- The angle that sweeps an arc equal to the radius on a circle. A full circle is 2 pi radians.
- Gradian (gon)
- An angle unit equal to one 400th of a full circle, used in some surveying. A right angle is 100 gradians.
- Arcminute
- One sixtieth of a degree, used for small angles in astronomy and navigation.
- Arcsecond
- One sixtieth of an arcminute, or one 3600th of a degree, for very small angles.
- Revolution
- One full turn, equal to 360 degrees or 2 pi radians. Also called a turn.
Frequently asked questions
How do I convert degrees to radians?
Multiply the degree value by pi and divide by 180. For example, 90 degrees × pi ÷ 180 = pi ÷ 2, which is about 1.5708 radians. This converter does it instantly when you type in the Degrees box.
How do I convert radians to degrees?
Multiply the radian value by 180 and divide by pi. For example, 1 radian × 180 ÷ pi is about 57.2958 degrees. Type a value into the Radians box and read the Degrees box.
How many radians are in a full circle?
A full circle is 2 pi radians, which is about 6.2832 radians. That same turn is 360 degrees, 400 gradians or 1 revolution.
What is a gradian and how does it relate to degrees?
A gradian (or gon) divides a full circle into 400 equal parts, so a right angle is 100 gradians. One degree equals 10 ÷ 9, or about 1.1111, gradians.
What is the difference between an arcminute and an arcsecond?
An arcminute is one sixtieth of a degree and an arcsecond is one sixtieth of an arcminute, so there are 60 arcseconds in an arcminute and 3600 arcseconds in a degree.
Why do calculators and code usually use radians?
Radians are the natural mathematical unit: calculus formulas, series expansions and physics equations for angles are simplest when angles are in radians. That is why functions like sine and cosine in most languages expect radian input.