Radian

1.1 Radian
 
Mind map titled 'Radian' illustrating the formula for converting radians to degrees.
 
Definition of One Radian
Definition

The angle subtended at the centre of a circle is \(1\) radian if the length of the arc is equal to the radius of the circle.

Figure

A triangle representing 1 radian where the radius matches the arc length.

 
Relationship with Degrees
  • \(1\text{ rad}=\dfrac{180^\circ}{\pi}\)
  • \(360^\circ=2\pi \text{ rad}\)
 
Conversion Formula
  • To convert degrees to radians:
    \(\text{radians}=\text{degrees}\times \dfrac{\pi}{180^\circ}\)
  • To convert radians to degrees:
    \(\text{degrees}=\text{radians}\times\dfrac{180^\circ}{\pi}\)
 
Example \(1\)
Question

Convert \(120^\circ\) to radians.

Solution

\(\begin{aligned} \text{radians}&=120^\circ\times\dfrac{\pi}{180^\circ} \\\\ &=\dfrac{2\pi}{3}\text{ rad} .\end{aligned}\)

 
Example \(2\)
Question

Convert \(\dfrac{\pi}{4}\text{ rad}\) to degrees.

Solution

\(\begin{aligned} \text{degrees}&=\dfrac{\pi}{4}\text{ rad}\times\dfrac{180^\circ}{\pi} \\\\ &=45^\circ. \end{aligned}\)

 

Radian

1.1 Radian
 
Mind map titled 'Radian' illustrating the formula for converting radians to degrees.
 
Definition of One Radian
Definition

The angle subtended at the centre of a circle is \(1\) radian if the length of the arc is equal to the radius of the circle.

Figure

A triangle representing 1 radian where the radius matches the arc length.

 
Relationship with Degrees
  • \(1\text{ rad}=\dfrac{180^\circ}{\pi}\)
  • \(360^\circ=2\pi \text{ rad}\)
 
Conversion Formula
  • To convert degrees to radians:
    \(\text{radians}=\text{degrees}\times \dfrac{\pi}{180^\circ}\)
  • To convert radians to degrees:
    \(\text{degrees}=\text{radians}\times\dfrac{180^\circ}{\pi}\)
 
Example \(1\)
Question

Convert \(120^\circ\) to radians.

Solution

\(\begin{aligned} \text{radians}&=120^\circ\times\dfrac{\pi}{180^\circ} \\\\ &=\dfrac{2\pi}{3}\text{ rad} .\end{aligned}\)

 
Example \(2\)
Question

Convert \(\dfrac{\pi}{4}\text{ rad}\) to degrees.

Solution

\(\begin{aligned} \text{degrees}&=\dfrac{\pi}{4}\text{ rad}\times\dfrac{180^\circ}{\pi} \\\\ &=45^\circ. \end{aligned}\)