# Conversion of Radians into Degrees and Vice Versa

We know that the circumference of a circle of radius $r$ is $2\pi r = \left( l \right)$ and the angle formed by one complete revolution is $\theta$ radian; therefore, $\begin{gathered} \theta = \frac{l}{r} \\ \Rightarrow \theta = \frac{{2\pi r}}{r} \\ \Rightarrow \theta = 2\pi \,radian \\ \end{gathered}$

Thus we have the relationship

$\begin{gathered} 2\pi \,radian\, = 360^\circ \\ \Rightarrow \pi \,radian = 180^\circ \\ \Rightarrow 1{\text{ }}radian = \frac{{180^\circ }}{\pi }\,\,\, = \frac{{180^\circ }}{{3.1416}}\,\,\,\, = \,\,\,57.296^\circ \\ \end{gathered}$

Further
$\begin{gathered} 1^\circ = \frac{\pi }{{180}}\,radian \\ 1^\circ = \frac{{3.1416}}{{180}}\,\,\,\,\, = 0.0175\,radian \\ \end{gathered}$

Example:

Convert the following angles into degrees:

(i) $\frac{{2\pi }}{3}$ radians
(ii) $3 radians$

Solution:

(i)
$\begin{gathered} \frac{{2\pi }}{3}\,radians = \frac{2}{3}\left( {\pi \,radian} \right) \\ \frac{{2\pi }}{3}\,radians = \frac{2}{3}\left( {180^\circ } \right)\,\,\,\, = 120^\circ \\ \end{gathered}$

(ii)

$3{\text{ }}radians\; = 3{\text{ }}\left( {1{\text{ }}radian} \right) = {\text{ }}3\left( {57.296^\circ } \right)\,\,\,\, = 171.888^\circ$

Example:

Convert $54^\circ \,\,45’$ into radians.

Solution:

$\begin{gathered} 54^\circ \,\,45′ = \left( {54\frac{{45}}{{60}}} \right)^\circ = \left( {54\frac{3}{4}} \right)^\circ \,\,\, = \frac{{219^\circ }}{4} \\ \Rightarrow 54^\circ \,\,45′ = \frac{{219}}{4}\left( {1^\circ } \right) \\ \Rightarrow 54^\circ \,\,45′ = \frac{{219}}{4}\left( {0.0175} \right) = 54^\circ \,\,45′ = 0.958\,radians \\ \end{gathered}$

Most calculators automatically would convert degrees into radians and radians into degrees.

Example:

An arc subtends an angle of $70^\circ$ at the center of a circle and its length is $132mm$. Find the radius of the circle.

Solution:

$70^\circ \,\,\, = 70 \times \frac{\pi }{{180}} = \frac{{70}}{{180}}\left( {\frac{{22}}{7}} \right) = \frac{{11}}{9}\,\,\,\,\,\,\,\,\,\,\left( {\because \pi = \frac{{22}}{7}} \right)$
$\therefore \theta = \frac{{11}}{9}\,radian$ and $l = 132m.m$
$\therefore \theta = \frac{1}{r} \Rightarrow r = \frac{l}{\theta } = 132 \times \frac{9}{{11}} = 108m.m$

Example:

Find the length of the equatorial arc subtending an angle of $1^\circ$ at the center of the Earth, taking the radius of the Earth as 5400km.

Solution:

$1^\circ = \frac{\pi }{{180}} = \frac{{3.1416}}{{180}}\,radians$

$\therefore \theta = \frac{{3.1416}}{{180}}$ and $r = 6400\,km$
Now
$\theta = \frac{l}{r}$
$\Rightarrow l = r\theta = 6400 \times \frac{{31416}}{{1800000}} = 111.7\,km$