Numeracy, Maths and Statistics - Academic Skills Kit

Radians

ContentsToggle Main Menu 1 Definition 2 Worked Examples 3 Video Example 4 Workbook 5 External Resources

Definition

Angles can be measured in units of degrees or radians. A complete revolution is defined as $360^\circ$ which is equal to $2\pi$ radians

$360^\circ = 2\pi \text{ radians.}$

From this, we can derive that

$1^{\circ} = \frac{\pi}{180} \text{ radians.}$ $1 \text{ radian} = \frac{180}{\pi}^{\circ}$

Sometimes a superscript ^c is used to denote radians instead of degrees, though it is conventional to assume that radians are used unless otherwise specified.

Worked Examples

Example 1

Convert $83^{\circ}$ to radians.

Solution

Recall that $1^{\circ} = \dfrac{\pi}{180} \text{ radians}$ . So multiply $\dfrac{\pi}{180}$ by $83$ :

$\begin{align} 83^{\circ} &= 83 \times \frac{\pi}{180}\\ &\approx 1.449 \text{ radians (to 3 d.p.)} \end{align}$

Example 2

Convert $3 \text{ radians}$ into degrees.

Solution

Using the definition, if $1 \text{ radian} = \dfrac{180}{\pi} \text{ degrees}$ , then multiply by $3$ to find the angle in degrees.

$3 \times \frac{180}{\pi} \approx 172^{\circ} \text{ (to 3 sig.fig.)}$

Video Example

Prof. Robin Johnson shows how to convert $37^{\circ}$ to radians, and $1$ radian to degrees.

Workbook

This workbook produced by HELM is a good revision aid, containing key points for revision and many worked examples.

Trigonometric functions including work on radians.

External Resources

Degrees and radians workbook at mathcentre.