Degrees to Radians

We define one radian (rad) as the central angle of a circle that subtends an arc equal to the length of the radius of the circle.

As you can see, one radian is quite a bit larger than one degree. In fact, one radian is approximately 57.3°.  An angle measuring two radians subtends an arc of length two times the radius of the circle, three radians subtends an arc of length 3 times the radius, and so on.  In general, the arc length can be expressed as some factor times the radius follows.

\[ \begin{align}
  s &= \theta  \cdot r \hfill \\
  \frac{{\,s\,}}{r}& = \theta  \hfill \\
\end{align} \]
It is this factor θ, the ratio of the arc length s subtended by a central angle divided by the radius r of the circle, which defines radian measure of an angle.

 

For example, consider the circle of radius 2 centimeters in which central angle q subtends an arc of length 6 centimeters as illustrated below.

The radian measure of the angle is,

\[ \theta  = \frac{s}{r} = \frac{{6\,{\rm{cm}}}}{{2\,{\rm{cm}}}} = 3\]

Notice that the definition results in a unitless real number.  When units are preferred, we will use rad as an abbreviation for radians, however, this is not required. Moving forward, always consider values for angles without expressed units to be radian measure.

\(\theta  = {3^\circ }\)  Three degrees.

\(\left. \begin{array}{l}\theta  = 3\\\theta  = 3{\mkern 1mu} {\mkern 1mu} rad{\mkern 1mu} \end{array} \right\}\) Three radians.

Certainly, angles measuring 3° and 3 radians are two very different angles.

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Degree to radian converter was created using MathJax and the Dart language.