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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Convert using the following facts.

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