Degrees and Radians

What a radian actually is

A degree is an arbitrary unit: 360 of them make a full circle, likely borrowed from the 360 days of an ancient calendar year. A radian, the SI unit for angles, is defined geometrically instead: it's the angle you sweep out when the arc length traveled equals the radius:

\text{Radian} = \dfrac{\text{arc length}}{\text{radius}}

A full circle's circumference is $2\pi r$ (exactly $2\pi$ radius-lengths around), so a full $360^\circ$ turn is exactly $2\pi$ radians.

Standard-angle conversions

0^\circ=0, \quad 30^\circ=\dfrac{\pi}{6}, \quad 45^\circ=\dfrac{\pi}{4}, \quad 60^\circ=\dfrac{\pi}{3}, \quad 90^\circ=\dfrac{\pi}{2}, \quad 180^\circ=\pi

Worked example: degrees to radians

Convert $150^\circ$ to radians.

Solution: Multiply by $\tfrac{\pi}{180}$ :

150 \times \dfrac{\pi}{180} = \dfrac{5\pi}{6}

Worked example: radians to degrees

Convert $\dfrac{5\pi}{4}$ radians to degrees.

Solution: Multiply by $\tfrac{180}{\pi}$ :

\dfrac{5\pi}{4} \times \dfrac{180}{\pi} = 5 \times 45 = 225^\circ