Topic 15 Circular measure

15.1 Radians and degrees

Angles are normally measured in degrees, e.g. a right angle has $90^\circ$. In mathematics and engineering it is more natural and convenient to measure angles in radians: the length of an arc of a unit circle is equal to the measurement, in radians, of the angle that it subtends.

Figure 15.1: Unit circle and radians

degrees	$360$	$180$	$90$	$60$	$45$	$30$	$0$
radians	$2\pi$	$\pi$	$\frac{\pi}{2}$	$\frac{\pi}{3}$	$\frac{\pi}{4}$	$\frac{\pi}{6}$	$0$

15.1.1 Converting degrees to radians

$$\theta^\circ = \left(\dfrac{2\pi}{360}\cdot\theta\right)\,\mathrm{rad}$$

Example 15.1 Find the value of $23^\circ$ in radians.

Solution. $23^\circ = \left(\dfrac{2\pi}{360}\cdot 23\right)\,\mathrm{rad} \approx 0.4\,\mathrm{rad}$.

15.1.2 Converting radians to degrees

$$\theta\,\mathrm{rad}=\left(\dfrac{360}{2\pi}\cdot\theta\right)^\circ$$

Example 15.2 Find the value of $1.5\,\mathrm{rad}$ in degrees.

Solution. $1.5\,\mathrm{rad} = \left(\dfrac{360}{2\pi}\cdot 1.5\right)^\circ = \left(\dfrac{270}{\pi}\right)^\circ \approx 85.9^\circ$.

15.2 Arc and sector

15.2.1 Length of an arc

The length of an arc of a circle is given by $s=r\cdot\theta$, where $r$ is the radius and $\theta$ is the angle subtended in radians.

Example 15.3 Determine the length $s$ of an arc of a circle with radius $45mm$ when the angle subtended is $85^\circ$.

Solution. $s=r\theta = 45\,\mathrm{mm}\cdot\left(\dfrac{2\pi}{360}\cdot 85\right) \approx 66.8\,\mathrm{mm}$.

15.2.2 Area of a sector

The area of a sector of a disc is $A=\left(\dfrac{\theta}{2\pi}\cdot\pi\cdot r^2\right) = \dfrac{1}{2}\cdot r^2\cdot \theta$, where $r$ is the radius of $\theta$ is the angle subtended in radians.

Example 15.4 A light source spreads illumination through an angle $130^\circ$ to a distance of $35m$. Determine the illuminated area $A$.

Solution. $A=\frac12 r^2\theta = \frac12\left(35\,\mathrm{m}\right)^2\cdot\left(\dfrac{2\pi}{360}\cdot 130\right) \approx 1390\,\mathrm{m}^2$.