Radian Conversion Calculator
Enter a value in radians below to convert to another unit of angle.
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Radian Conversion Calculators
You can also convert angle using one of our radian converters below.
What Is a Radian?
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A radian is a unit used to measure angle.
The radian is the SI derived unit for angle in the metric system. Radians can be abbreviated as rad, and are also sometimes abbreviated as c, r, or R. For example, 1 radian can be written as 1 rad, 1 c, 1 r, or 1 R.
Radians are often expressed using their definition. The formula to find an angle in radians is θ = s/r, where the angle in radians θ is equal to the arc length s divided by the radius r. Thus, radians may also be expressed as the formula of arc length over the radius.
Radians are also considered to be a "unitless" unit. That is, when multiplying or dividing by radians, the result does not include radians as part of the final units.
For example, when determining the length of an arc for a given angle, we use the formula above, rearranged to be s = θr. If θ is in radians and r is in meters, then the units of s will be meters, not radian-meters. If θ were in degrees, however, then s would have units of degree-meters.
Background and Origin
The radian was designed in 1714 by Roger Cotes, but it wasn’t named and defined in publication until 1873. It was designed as a more natural measure of angle as an alternative to the degree.
Uses
Radians are the defacto unit for measuring angles in calculus and physics. Radians are more easily expressed in formulae and are a more natural mathematical unit, hence the appeal in these fields.
How Much Is a Radian?
A radian is the measurement of angle equal to the length of an arc divided by the radius of the circle or arc.[1] 1 radian is equal to 180/π degrees, or about 57.29578°. There are about 6.28318 radians in a circle.
How to Convert Radians
To convert radians to another unit of angle, you need to multiply the value by a conversion factor. A conversion factor is a numerical value used to change the units of a measurement without changing the value.[2]
You can find the conversion factors for radians in the conversion table below.
Then, multiply the angle measurement by the conversion factor to find the equivalent value in the desired unit of measurement.
radians × conversion factor = result
You can also use a calculator, such as one of the converters below, for the conversion.
Radian Conversion Table
| radians | milliradians | degrees | minutes of arc | seconds of arc | gradians | revolutions | circles | Mils (NATO) |
|---|---|---|---|---|---|---|---|---|
| 1 rad | 1,000 mrad | 57.29578° | 3,438' | 206,265" | 63.661977g | 0.159155 r | 0.159155 cir | 1,019 mil |
| 2 rad | 2,000 mrad | 114.591559° | 6,875' | 412,530" | 127.323954g | 0.31831 r | 0.31831 cir | 2,037 mil |
| 3 rad | 3,000 mrad | 171.887339° | 10,313' | 618,794" | 190.985932g | 0.477465 r | 0.477465 cir | 3,056 mil |
| 4 rad | 4,000 mrad | 229.183118° | 13,751' | 825,059" | 254.647909g | 0.63662 r | 0.63662 cir | 4,074 mil |
| 5 rad | 5,000 mrad | 286.478898° | 17,189' | 1,031,324" | 318.309886g | 0.795775 r | 0.795775 cir | 5,093 mil |
| 6 rad | 6,000 mrad | 343.774677° | 20,626' | 1,237,589" | 381.971863g | 0.95493 r | 0.95493 cir | 6,112 mil |
| 7 rad | 7,000 mrad | 401.070457° | 24,064' | 1,443,854" | 445.633841g | 1.114085 r | 1.114085 cir | 7,130 mil |
| 8 rad | 8,000 mrad | 458.366236° | 27,502' | 1,650,118" | 509.295818g | 1.27324 r | 1.27324 cir | 8,149 mil |
| 9 rad | 9,000 mrad | 515.662016° | 30,940' | 1,856,383" | 572.957795g | 1.432394 r | 1.432394 cir | 9,167 mil |
| 10 rad | 10,000 mrad | 572.957795° | 34,377' | 2,062,648" | 636.619772g | 1.591549 r | 1.591549 cir | 10,186 mil |
| 11 rad | 11,000 mrad | 630.253575° | 37,815' | 2,268,913" | 700.28175g | 1.750704 r | 1.750704 cir | 11,205 mil |
| 12 rad | 12,000 mrad | 687.549354° | 41,253' | 2,475,178" | 763.943727g | 1.909859 r | 1.909859 cir | 12,223 mil |
| 13 rad | 13,000 mrad | 744.845134° | 44,691' | 2,681,442" | 827.605704g | 2.069014 r | 2.069014 cir | 13,242 mil |
| 14 rad | 14,000 mrad | 802.140913° | 48,128' | 2,887,707" | 891.267681g | 2.228169 r | 2.228169 cir | 14,260 mil |
| 15 rad | 15,000 mrad | 859.436693° | 51,566' | 3,093,972" | 954.929659g | 2.387324 r | 2.387324 cir | 15,279 mil |
| 16 rad | 16,000 mrad | 916.732472° | 55,004' | 3,300,237" | 1,019g | 2.546479 r | 2.546479 cir | 16,297 mil |
| 17 rad | 17,000 mrad | 974.028252° | 58,442' | 3,506,502" | 1,082g | 2.705634 r | 2.705634 cir | 17,316 mil |
| 18 rad | 18,000 mrad | 1,031° | 61,879' | 3,712,767" | 1,146g | 2.864789 r | 2.864789 cir | 18,335 mil |
| 19 rad | 19,000 mrad | 1,089° | 65,317' | 3,919,031" | 1,210g | 3.023944 r | 3.023944 cir | 19,353 mil |
| 20 rad | 20,000 mrad | 1,146° | 68,755' | 4,125,296" | 1,273g | 3.183099 r | 3.183099 cir | 20,372 mil |
References
- International Bureau of Weights and Measures, The International System of Units, 9th Edition, 2019, https://www.bipm.org/documents/20126/41483022/SI-Brochure-9-EN.pdf
- National Institute of Standards & Technology, Unit Conversion, https://www.nist.gov/pml/owm/metric-si/unit-conversion