When dealing with circles and circular motion, it is more convenient to measure angles and angular displacements in units called radians rather than in degrees.
If an object moves a distance s around a circular path of radius r (Figure 16.3a), its angular displacement $\theta $ in radians is defined as follows:

$\theta  = \frac{s}{r}$

Since both s and r are distances measured in metres, it follows that the angle $\theta $ is simply a ratio. It is a dimensionless quantity. If the object moves twice as far around a circle of twice the radius (Figure 16.3b), its angular displacement $\theta $ will be the same.

$\begin{array}{l}
\theta  = \frac{{length\,of\,arc}}{{radius}}\\
 = \frac{{2s}}{{2r}}\\
 = \frac{s}{r}
\end{array}$

Figure 16.3: The size of an angle depends on the radius and the length of the arc. Doubling both leaves the angle unchanged.

When we define $\theta $ in this way, its units are radians rather than degrees. How are radians related to degrees? If an object moves all the way round the circumference of the circle, it moves a distance of $2\pi r$.
We can calculate its angular displacement in radians:

$\begin{array}{l}
\theta  = \frac{{circumference}}{{radius}}\\
 = \frac{{2\pi r}}{{2r}}\\
 = 2\pi 
\end{array}$

Hence a complete circle contains $2\pi $ radians. But we can also say that the object has moved through ${360^ \circ }$.
Hence:

${360^ \circ } = 2\pi \,rad$

Similarly, we have:
$\begin{array}{l}
{180^ \circ } = \pi \,rad\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{90^ \circ } = \frac{\pi }{2}\,rad\\
{45^ \circ } = \frac{\pi }{2}\,rad
\end{array}$ 
and so on

Defining the radian

One radian is defined as the angle subtended at the centre of a circle by an arc of length equal to the radius of the circle.
This is illustrated in Figure 16.4.

An angle of ${360^ \circ }$ is equivalent to an angle of $2\pi $ radians. We can therefore determine what 1 radian is equivalent to in degrees.

$1\,radian = \frac{{{{360}^ \circ }}}{{2\pi }}$
or $1\,radian \approx {57.3^ \circ }$

If you can remember that there are $2\pi $ rad in a full circle, you will be able to convert between radians and degrees:
- to convert from degrees to radians, multiply by $\frac{{2\pi }}{{{{360}^ \circ }}}$ or $\frac{{\pi }}{{{{180}^ \circ }}}$.
- to convert from degrees to degrees, multiply by $\frac{{{{360}^ \circ }}}{{2\pi }}$ or $\frac{{{{180}^ \circ }}}{{\pi }}$.
Now look at Worked example 1.