The radian measure of an angle is the ratio of the length of the arc (s) whose centre is at the vertex of the angle to the radius (r).
$\displaystyle \begin{array}{{|l|}}\hline \displaystyle \theta =\frac{s}{r} \\ \hline \end{array}$
For one complete anticlockwise revolution, the length of the arc is the length of entire circumference and hence $ \displaystyle s=2\pi r$. Therefore, the radian measure of one complete revolution is
In degrees, one complete counterclockwise revolution is $\displaystyle 360°$ and hence $ \displaystyle \theta=360°$.
$ \displaystyle \begin{array}{l}\therefore \ \ 360{}^\circ =2\pi \ \text{radians}\\\\\therefore \ \ 180{}^\circ =\pi \ \text{radians}\\\\\therefore \ \ 1{}^\circ \times 180=\pi \ \text{radians}\\\end{array}$
$\displaystyle \therefore\ $ $\displaystyle \begin{array}{{|l|}}\hline \displaystyle 1{}^\circ =\frac{\pi }{{180}}\ \text{radians} \\ \hline \end{array}$
$ \displaystyle \begin{array}{l}\ \ \ \ \text{Since }\pi \ \text{radians}=180{}^\circ ,\\\\\ \ \ \ 1\ \text{radian}\ \times \pi =180{}^\circ \end{array}$
$\displaystyle \therefore\ $$\displaystyle \begin{array}{{|l|}}\hline \displaystyle 1\ \text{radian}\ =\frac{{180{}^\circ }}{\pi } \\ \hline \end{array}$
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