Angle Unit Converter: Degrees, Radians, Gradians, Turns
Convert an angle between degrees, radians, gradians (gon) and turns, all four at once. One turn is exactly 360 degrees and 400 gon, and it is 2π radians.
Angle Converter
Type an angle in one unit and read it in turns, degrees, gradians and radians at once.
Converted angle
Documentation
Angle Unit Converter
An angle measures how far one direction has been turned away from another. An angle unit converter rewrites an angle from one unit into another, such as degrees into radians. This page converts between turns, degrees, gradians (also written gon) and radians, and shows all four values at the same time.
What the converter does
A visitor types one number and picks the unit that number is written in. The page returns the same angle in all four units.
- Results are shown to eight significant figures. Only the display is rounded. The conversion runs on the number as typed.
- A result smaller than 0.0001, or of a billion or more, is shown in scientific notation. Zero is shown as a plain 0.
- Negative angles are converted, because a rotation has a direction.
- Angles are not folded into a single revolution. 720 degrees comes out as 2 turns, not as 0.
- An empty box gives a prompt rather than a number. A link carrying a unit this page does not offer gives a message rather than a conversion.
The four units
| Unit | Symbol | Turns in one unit | Exact by definition |
|---|---|---|---|
| Turn | turn | 1 | Yes, one full revolution |
| Degree | ° | 1/360 | Yes |
| Gradian (gon) | gon | 1/400 | Yes |
| Radian | rad | 1/(2π) | Yes as a fraction, no as a decimal |
A turn is one complete revolution, back to the direction it started from. The degree cuts that revolution into 360 parts, a habit inherited from Babylonian counting in sixties. The gradian cuts it into 400 parts, which makes a right angle a round 100. The radian is the angle whose arc along a circle is as long as that circle's radius, and 2π of them fit in a turn.
Angle conversion formula
Each unit is a fixed number of turns, so a conversion is one multiplication followed by one division. The angle is carried into turns and then back out into the unit that is wanted:
where is the angle typed in, is the number of turns in one unit of the chosen unit, is the number of turns in one unit of the wanted unit, and is the answer.
Nothing is added or taken away along the way. All four scales start counting from the same place, so only the size of the step changes. Temperature is the case where that is untrue: Celsius and Fahrenheit begin at different points, so a converter that only multiplied would report 0 °C as 0 °F instead of 32 °F.
Putting the degree and radian factors into that formula gives the pair people look up most often:
\qquad \theta_{\mathrm{deg}} = \theta_{\mathrm{rad}} \times \frac{180}{\pi}$$ where $\theta_{\mathrm{deg}}$ is the angle in degrees and $\theta_{\mathrm{rad}}$ is the same angle in radians. ### Why the page counts from turns and not from radians The turn is the unit the other three are measured against. That choice is not cosmetic. Degrees, gradians and turns are whole-number fractions of each other, so a computer can hold their sizes without error and give back answers that land exactly on the round number a reader expects. Radians cannot do that, because π has no exact decimal form. If the radian were the base unit, every degree and every gradian would have to travel through π and back. A right angle typed as 90 degrees would then return 99.99999999999999 gon rather than 100, and 200 gon would return 180.00000000000003 degrees. Counting from the turn keeps π in the one row that genuinely needs it. ### Worked example: 100 degrees 100 degrees is the angle the page starts with. One degree is 1/360 of a turn, so the page multiplies 100 by 1/360 and gets 0.2777777777777778 turn. It shows that as 0.27777778. The other two answers come from dividing that value by their own factors. Dividing by 1/400 is the same as multiplying by 400, which gives 111.11111111111111 gon, shown as 111.11111. Dividing by 1/(2π) gives 1.7453292519943295 rad, shown as 1.7453293. A right angle is tidier. 90 degrees is a quarter of a turn, so the page returns 0.25 turn, exactly 100 gon, and 1.5707963 rad. ### Common angle conversions Each row shows what one value produces in all four units, at eight significant figures. | Input | turn | degrees | gon | radians | |---|---|---|---|---| | 1 turn | 1 | 360 | 400 | 6.2831853 | | 1 degree | 0.0027777778 | 1 | 1.1111111 | 0.017453293 | | 1 gon | 0.0025 | 0.9 | 1 | 0.015707963 | | 1 radian | 0.15915494 | 57.29578 | 63.661977 | 1 | | 90 degrees | 0.25 | 90 | 100 | 1.5707963 | | 180 degrees | 0.5 | 180 | 200 | 3.1415927 | ### Which values are exact and which are rounded Three of the four factors are exact fractions, so a conversion among turns, degrees and gradians is limited only by the arithmetic, never by the constants. - One turn is exactly 360 degrees. The degree is not an SI unit, but the International Bureau of Weights and Measures lists it in the SI Brochure among the non-SI units accepted for use with the SI, with one degree equal to π/180 radians. - One turn is exactly 400 gradians. ISO 80000-3, the standard covering quantities and units of space and time, gives one gon as π/200 radians, which is a four-hundredth of a turn. - One turn is 2π radians. The radian is the coherent SI unit of plane angle, defined as the ratio of an arc to the radius, so this relation is a definition as well. π is irrational, though: its decimal never ends and never repeats, so no written decimal for it is the whole number. - The radian factor is therefore rounded in practice. The page builds it from the value of π a computer carries, 3.141592653589793, which is the closest a double-precision number can get. Radian answers inherit that rounding. The other three units do not. The rounding a reader actually sees comes from the display rather than from the constants. Eight significant figures is wide enough to show a defining value in full and narrow enough to hide the last digits of the computer's arithmetic, which carry rounding noise rather than information. ### Frequently asked questions **How many degrees is one radian?** 57.29578 degrees at eight significant figures, usually quoted as 57.3. Written exactly it is 180/π degrees, and π has no exact decimal. **How many radians is 90 degrees?** 1.5707963 rad at eight significant figures. Written exactly it is π/2 radians. **Why is a right angle 100 gon?** Because a full turn holds 400 gradians and a right angle is a quarter of a turn. The gradian was designed for that, so surveyors could work in a decimal system instead of in 90 parts. **Is 1 gon the same as 0.9 degrees?** Yes, exactly. A turn is 400 gon and also 360 degrees, so one gon is 360/400 of a degree. **Does the page reduce 720 degrees to 0?** No. It converts the number it is given, so 720 degrees returns 2 turns, 800 gon and 12.566371 rad. Folding angles into one revolution would throw away how many times something had turned. **Can a negative angle be converted?** Yes. A negative value means the rotation runs the other way, so it is converted like any other number. Typing -45 degrees returns -0.125 turn, -50 gon and -0.78539816 rad.