Conversions are performed by using a conversion factor. By knowing the conversion factor, converting between units can become a simple multiplication problem:
S * C = E
Where S is our starting value, C is our conversion factor, and E is our end converted result.
To simply convert from any unit into degrees, for example, from 5 radians, just multiply by the value in the right column in the table below.
5 radians * 57.29578 [degrees / radian] = 286.4789 degrees
To convert from degrees back into units in the left column divide by the value in the right column or, multiply by the reciprocal, 1/x.
286.4789 degrees / 57.29578 [degrees / radian] = 5 radians
To convert among any units in the left column, say from A to B, you can multiply by the factor for A to convert A into degrees then divide by the factor for B to convert out of degrees. Or, you can find the single factor you need by dividing the A factor by the B factor.
For example, to convert from radians to revolutions you would multiply by 57.29578 then divide by 360. Or, multiply by 57.29578/360 = 0.1591549. So, to convert directly from radians to revolutions you multiply by 0.1591549.
To understand how to also convert the units follow this example. Say you want to convert from radians to degrees. Since you can multiply anything by 1 and still retain the original value, but in different units, set it up so that radian will cancel out leaving you with degree.
1 degree = 0.01745329 radians, 1 degree / 0.01745329 radians = 1
We can write the conversion as:
1 radian = 1 radian * (1 degree / 0.01745329 radians) = 57.29578 degrees
And we now have our factor for conversion from radians to degrees since 1 * 57.29578 = 57.29578. Note that there are rounding errors in these values.
Knowing that 1 radian = 57.29578 degrees we can now find the conversion factor for converting back. Dividing both sides of the equation by 57.29578 we get about 0.01745329 radians = 1 degrees. So, the conversion factor to multiply by to convert from degrees to radians is about 0.01745329.
( ° )
(degrees / Unit)
( rad )
(radians / Unit)
The National Institute of Standards and Technology (NIST) - The NIST Guide for the use of the International System of Units - Appendix B, subsections B.8 Factors for Units Listed Alphabetically and B.9 Factors for units listed by kind of quantity or field of science.
Lide, David R., Daniel (Editor-in-Chief). CRC Handbook of Chemistry and Physics, 89th Edition New York, NY: CRC Press, p. 1-28, 2008.
Wikipedia contributors. "Conversion of units" Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, last visited 26 Jun. 2011.