Radian

1.1

Radian

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

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

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

1.1

Radian

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

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

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