Radians
Radians are a natural way to measure angles using the relationship between the length of an arc and the radius of a circle. Unlike degrees, the radian measure is obtained directly from the geometry of the circle.
Definition of radian
An angle measures one radian when it subtends an arc whose length is equal to the radius of the circle.
$$\theta=\frac{s}{r}$$
Here, \(s\) is the arc length, \(r\) is the radius, and \(\theta\) is the angle measured in radians. Therefore, when \(s=r\), we have \(\theta=1\) radian.
Activity: Interactive chart
Move the angle in the GeoGebra construction and compare the radius of the circle with the length of the corresponding arc. Observe that one radian is obtained when the arc length is exactly one radius.
Degrees and radians
A complete turn has \(360^\circ\). The circumference of a circle is \(2\pi r\), so a complete turn also measures \(2\pi\) radians.
$$360^\circ=2\pi\ \mathrm{rad} \qquad\Longrightarrow\qquad 180^\circ=\pi\ \mathrm{rad}$$
To convert degrees to radians, multiply by \(\pi/180\).
$$\theta_{\mathrm{rad}}=\theta_{\mathrm{deg}}\frac{\pi}{180}$$
To convert radians to degrees, multiply by \(180/\pi\).
$$\theta_{\mathrm{deg}}=\theta_{\mathrm{rad}}\frac{180}{\pi}$$
Notable angles
The following equivalences are especially useful in trigonometry and calculus.
| Degrees | Radians |
|---|---|
| \(0^\circ\) | \(0\) |
| \(30^\circ\) | \(\frac{\pi}{6}\) |
| \(45^\circ\) | \(\frac{\pi}{4}\) |
| \(60^\circ\) | \(\frac{\pi}{3}\) |
| \(90^\circ\) | \(\frac{\pi}{2}\) |
| \(180^\circ\) | \(\pi\) |
| \(270^\circ\) | \(\frac{3\pi}{2}\) |
| \(360^\circ\) | \(2\pi\) |
Arc length
Because radian measure is defined by \(\theta=s/r\), the length of an arc can be calculated directly when the angle is expressed in radians.
$$s=r\theta$$
For example, if a circle has radius \(5\) and the central angle is \(\pi/3\) radians, then the arc length is \(s=5\pi/3\).