Introduction:
Up to this stage, you have been using degrees to measure the sizes of angles. However, for a variety of reasons, angle measures in advanced mathematics are frequently described using a unit system different from the degreesystem. This system is known as the radian system.
What is a Radian?
The radian is the S.I. unit for measuring angles and is the standard unit of angular measure used in many areas of mathematics. The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian is just under 57.3 degrees.
To understand what a radian is, consider the following figure, which shows two arcs of two concentric circles subtending the same angle \(\theta \) at their center O.
Let the lengths of the two arcs be \({l_1}\) and \({l_2}\), and let the radii of the corresponding circles be \({r_1}\) and \({r_2}\).
The arcs are similar, because they subtend the same angle at the center of the circle of which they are a part. Intuitively, this indicates that the arc length to radius ratio of the arcs must be the same:
\[\frac{{{l_1}}}{{{r_1}}} = \frac{{{l_2}}}{{{r_2}}}\]
This can be mathematically verified, as we know how to calculate the arc length, given the angle subtended at the center:
\[\begin{align}&{l_1} = \frac{\theta }{{{{360}^0}}} \times 2\pi {r_1}\,\,\, \Rightarrow \,\,\,\frac{{{l_1}}}{{{r_1}}} = \frac{\theta }{{{{360}^0}}} \times 2\pi \\&{l_2} = \frac{\theta }{{{{360}^0}}} \times 2\pi {r_2}\,\,\, \Rightarrow \,\,\,\frac{{{l_2}}}{{{r_2}}} = \frac{\theta }{{{{360}^0}}} \times 2\pi \end{align}\]
Thus, the \(l/r\) ratio for an arc is a measure of the angle it subtends at the center. The larger is this ratio, the larger is the angle. This means that rather than measuring angles in degrees, we can use the \(l/r\) ratio of measure the sizes of angles.
Let us see how. Consider an angle of 360^{0}, which is a complete angle:
The arc length \(l\), in this case, is the entire circumference of the circle, or \(l = 2\pi r\). Thus, the \(l/r\) ratio for an angle of 360^{0} is \(2\pi \). We will say that 360^{0} is equal to \(2\pi \) radians – this should be interpreted as saying that 360^{0} corresponds to \(l/r\) ratio of \(2\pi \).
✍Note: \(2\pi \) radians \( = 360^\circ \)
Next, consider an angle of 180^{0}:
The arc length \(l\) in this case corresponds to half of the circumference, or \(l = \pi r\). Thus, the \(l/r\) ratio in this case is equal to \(\pi \). We will say that 180^{0} equals \(\pi \) radians.
✍Note: \(\pi \) radians \( = 180^\circ \)
As another example, consider an angle of 90^{0}, which corresponds to an arc length of onefourth of the circumference:
We have \(l = \frac{1}{2}\pi r\), so that \(l/r\) ratio in this case equals \(\frac{\pi }{2}\). Thus, 90^{0} equals \(\frac{\pi }{2}\) radians.
Till now, we have seen that

\(2\pi \) radians \( = 360^\circ \)

\(\pi \) radians \( = 180^\circ \)

\(\frac{\pi }{2}\) radians \( = 90^\circ \)
How do we define 1 radian now? When the arc length is exactly equal to the radius, so that the \(l/r\) ratio is exactly 1, we have an angle subtended of 1 radian:
✍Note: 1 radian is the angle made when the radius is wrapped round the circle.
What is the size of 1 radian in degrees? Well, since \(\pi \) radians equal 180 degrees (as we have already seen),
\[1\,{\rm{radian}} = {\left( {\frac{{180}}{\pi }} \right)^0} \approx {57.3^0}\]
We can also conclude one more important result which can be used in calculations,
\[1^\circ = \frac{\pi }{{180}}{\text{ radians}}\]
Solved Examples:
Example 1: Write the measures of \({30^0},{\rm{ }}{45^0},{\rm{ }}{60^0}\;and\;{\rm{ }}{90^0}\) using the radian scale.
Solution: We have,
\[\begin{align}&{30^0} = \frac{{{{180}^0}}}{6} \equiv \frac{\pi }{6}\,\,\,{\rm{rad}}\\&{45^0} = \frac{{{{180}^0}}}{4} \equiv \frac{\pi }{4}\,\,\,{\rm{rad}}\\&{60^0} = \frac{{{{180}^0}}}{3} \equiv \frac{\pi }{3}\,\,\,{\rm{rad}}\\&{90^0} = \frac{{{{180}^0}}}{2} \equiv \frac{\pi }{2}\,\,\,{\rm{rad}}\end{align}\]
Example 2: Convert the following degree measures into radians:
 \({80^0}\)
 \({120^0}\)
Solution: We know that \(1^\circ = \frac{\pi }{{180}}{\text{ radians}}\), so we have,
\[{80^0} = \left( {80 \times \frac{\pi }{{180}}} \right)rad\]
\[ \Rightarrow \boxed{{{80}^0} = \frac{{4\pi }}{9}\,\,\,rad}\]
Similarily,
\[{120^0} = \left( {120 \times \frac{\pi }{{180}}} \right)rad\]
\[ \Rightarrow \boxed{{{120}^0} = \frac{{2\pi }}{3}\,\,\,rad}\]
Challenge: Convert the following radians into degrees:
 \({\frac{{4\pi }}{3}\,\,\,rad}\)
 \({\frac{{5\pi }}{4}\,\,\,rad}\)
⚡Tip: \(\pi \) radians \( = 180^\circ \).