# 6.1.1 Angular Displacement

The radian, denoted by the symbol rad, is the SI unit of angles.

Consider an angle subtended by an arc of a circle. The magnitude of that angle θ, in radian, is simply the ratio of the arc length s to the radius r.

$\displaystyle \theta =\frac{s}{r}$

Based on the definition, 1 radian would be the angle subtended by a circular arc same length as the radius, which turns out to be about 60°. To be more exact, $1\text{ rad}=57.29577951{}^\circ$ (8 d.p.).

Since the circumference of a circle would subtend an angle of 360°, 360° must correspond to $\displaystyle \theta =\frac{s}{r}=\frac{{2\pi r}}{r}=2\pi \text{ rad}$. By simple proportion, we can work out the other “important” angles.

\displaystyle \begin{aligned}360{}^\circ &=2\pi \text{ rad}\\180{}^\circ &=\pi \text{ rad}\\90{}^\circ &=\frac{\pi }{2}\text{ rad}\\45{}^\circ &=\frac{\pi }{4}\text{ rad}\end{aligned}

Personally, I “memorise” $180{}^\circ =\pi \text{ rad}$ to help me convert any angle between ° and rad by using simple proportion. E.g.

$\displaystyle 30{}^\circ =\frac{{30{}^\circ }}{{180{}^\circ }}\times \pi =0.524\text{ rad}$

and

$\displaystyle 3.0\text{ rad}=\frac{{3.0}}{\pi }\times 180{}^\circ =172{}^\circ$.

Concept Test

1001