Angles are commonly measured in degrees but sometimes, it is useful to define them in terms of the length around the unit circle. This resource introduces radians as a measure of angle. Being able to use radians is crucial for solving problems in engineering, such as analysing the motion of gears and wheels, and in computer graphics, where they help in rendering angles and rotations accurately.
Radians
Angles can be measured in units known as radians. One radian is the angle created by bending the radius length around the arc of a unit circle. A unit circle has a radius of \(1\). You will learn more about the unit circle here.
Consider the circle centred at the origin. The red arc is the same length as the radius of the circle. The angle subtended at (or formed from) the origin \(\theta\) is one radian and is denoted by \(1^{c}\).
In the examples on this page, we show the unit symbol for radian (\(^{c}\)) but note that this is often omitted.
One radian is approximately \(57.3^{\circ}\).
Converting between degrees and radians
The circumference of a unit circle is given by the formula \(C=2\pi r\). Because of this, we know \(2\pi\) radians (or \(2\pi^{c}\)) is a complete rotation and the same as \(360\) degrees (or \(360^{\circ}\)). Similarly, half a rotation or \(180\) degrees \(=\pi\) radians (\(180^{\circ}=\pi^{c}\)).
We can express angles that represent fractional parts of a circle in terms of \(\pi\).
Example 1 – converting between degrees and radians
Convert \(60^{\circ}\) to radians.
Multiply the original quantity by the correct conversion factor. This is the conversion factor with the units we want in the numerator (top of the fraction) and the units we have in the denominator (bottom of the fraction).
We need to convert from degrees to radians, so the correct conversion factor is \(\dfrac{\pi^{c}}{180^{\circ}}\).
\[\begin{align*} 60^{\circ} & = 60^{\circ}\times\frac{\pi^{c}}{180}\\
& = \frac{60^{\circ}\pi^{c}}{180^{\circ}}\quad\textrm{cancel out degrees on top and bottom}\\
& = \frac{\pi^{c}}{3}\\
& \approx\frac{3.142^{c}}{3}\\
& \approx1.05^{c}
\end{align*}\]
Convert \(240^{\circ}\) to radians.
Multiply the original quantity by the correct conversion factor. This is the conversion factor with the units we want in the numerator (top of the fraction) and the units we have in the denominator (bottom of the fraction).
We need to convert from degrees to radians, so the correct conversion factor is \(\dfrac{\pi^{c}}{180^{\circ}}\).
\[\begin{align*} 240^{\circ} & = 240^{\circ}\times\frac{\pi^{c}}{180^{\circ}}\\
& = \frac{240^{\circ}\pi^{c}}{180^{\circ}}\quad\textrm{cancel out degrees on top and bottom}\\
& = \frac{4\pi^{c}}{3}
\end{align*}\]
Convert \(\dfrac{\pi}{4}\) radians to degrees.
Multiply the original quantity by the correct conversion factor. This is the conversion factor with the units we want in the numerator (top of the fraction) and the units we have in the denominator (bottom of the fraction).
We need to convert from radians to degrees, so the correct conversion factor is \(\dfrac{180^{\circ}}{\pi^{c}}\).
\[\begin{align*} \frac{\pi}{4}^{c} & = \frac{\pi}{4}^{c}\times\frac{180^{\circ}}{\pi^{c}}\quad\textrm{cancel out \(\pi\) radians from top and bottom}\\
& = \frac{180^{\circ}}{4}\\
& = 45^{\circ} \end{align*}\]
Convert \(6.5^{c}\) to degrees.
Multiply the original quantity by the correct conversion factor. This is the conversion factor with the units we want in the numerator (top of the fraction) and the units we have in the denominator (bottom of the fraction).
We need to convert from radians to degrees, so the correct conversion factor is \(\dfrac{180^{\circ}}{\pi^{c}}\).
\[\begin{align*} 6.5^{c} & = 6.5^{c}\times\frac{180^{\circ}}{\pi^{c}}\quad\textrm{cancel out radians from top and bottom}\\
& = \frac{1170^{\circ}}{\pi}\\
& \approx\frac{1170^{\circ}}{3.142}\\
& \approx372.4^{\circ}
\end{align*}\]
Your turn – converting between degrees and radians
