You now have two units for angles. Both describe a half turn: it is $180^\circ$ and it is $\pi$ radians (half of $2\pi$). That one fact is all you need to convert, as in radian measure.
Key idea
$$180^\circ = \pi \text{ radians}.$$ Degrees to radians: multiply by $\dfrac{\pi}{180^\circ}$. Radians to degrees: multiply by $\dfrac{180^\circ}{\pi}$.
Why it works
Since $180^\circ$ and $\pi$ radians are the same angle, the fraction $\frac{\pi}{180^\circ}$ equals $1$, and multiplying by $1$ never changes an amount. You only pick the version of $1$ that cancels the unit you have, the way you convert inches to feet.
| Degrees | $0^\circ$ | $30^\circ$ | $45^\circ$ | $60^\circ$ | $90^\circ$ | $120^\circ$ | $135^\circ$ | $150^\circ$ | $180^\circ$ | $270^\circ$ | $360^\circ$ |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Radians | $0$ | $\frac{\pi}{6}$ | $\frac{\pi}{4}$ | $\frac{\pi}{3}$ | $\frac{\pi}{2}$ | $\frac{2\pi}{3}$ | $\frac{3\pi}{4}$ | $\frac{5\pi}{6}$ | $\pi$ | $\frac{3\pi}{2}$ | $2\pi$ |
Worked examples
Worked example: Degrees to radians
Convert $135^\circ$ to radians.
- Multiply by $\dfrac{\pi}{180^\circ}$ so the degrees cancel: $135^\circ \cdot \dfrac{\pi}{180^\circ} = \dfrac{135\pi}{180}$.
- Simplify the fraction: $135$ and $180$ share a factor of $45$, so $\dfrac{135\pi}{180} = \dfrac{3\pi}{4}$.
Answer: $\frac{3\pi}{4}$
Worked example: Radians to degrees
Convert $\frac{7\pi}{6}$ to degrees.
- Multiply by $\dfrac{180^\circ}{\pi}$; the $\pi$ cancels.
- $\dfrac{7\pi}{6} \cdot \dfrac{180^\circ}{\pi} = \dfrac{7 \cdot 180^\circ}{6} = 7 \cdot 30^\circ$.
Answer: $210^\circ$
Worked example: No $\pi$ in sight
Convert $2$ radians to degrees.
- Same factor: $2 \cdot \dfrac{180^\circ}{\pi} = \dfrac{360^\circ}{\pi}$.
- That is about $114.6^\circ$, a bit more than a right angle.
Answer: $\frac{360^\circ}{\pi} \approx 114.6^\circ$
Common mistake
Multiplying by the wrong factor, for example $30^\circ \cdot \frac{180}{\pi} = \frac{5400}{\pi}$. Check the units: the unit you start with must cancel, so degrees go in the denominator when you convert to radians. Check the size too: a radian is about $57^\circ$, so the radian number is always much smaller than the degree number.