The radian is a unit of angular measure. One radian is defined to be the size of the angle at the center of a unit circle subtended by an arc of length one. It turns out that regardless of the size of the circle, such an angle is one radian whenever the radius is equal to the arc that subtends it.

In general, the magnitude in radians of a subtended angle at the center of a circle is equal to the ratio of the arc length to the radius:

    \[\theta = \dfrac{s}{r}\]

where \theta is the subtended angle, s is the length of the circular arc, and r is the radius.


Radian Diagram


This is useful because it means that if we know the measure of an angle in radians and the radius of a circular arc, r, we can calculate the length of the arc, s, by rearranging the formula to:

    \[s = r\theta\]

Since one complete revolution of a circle is 360º and the circumference of a circle is 2\pi{r}, it follows that there are \frac{2\pi{r}}{r}, or 2\pi, radians per 360º.

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