Convert between degrees and radians

Degree–radian conversion rests on the fact that a full rotation is $360^\circ=2\pi$ radians, so $180^\circ=\pi$ radians; measures are converted by multiplying by $\pi/180$ or $180/\pi$ . Radians represent the ratio of an arc’s length to the circle’s radius, supporting exact angle representations such as $30^\circ=\pi/6$ and $3\pi/4=135^\circ$ for trigonometric functions and graphs. Radian-based calculus formulas and more advanced applications are not included.

Detailed Explanation: Convert between degrees and radians

A full rotation measures $360^\circ$ or $2\pi$ radians. Therefore,

180^\circ=\pi\text{ radians}.

Use these conversion factors:

Degrees to radians: multiply by $\dfrac{\pi}{180}$
Radians to degrees: multiply by $\dfrac{180}{\pi}$

Example: Convert $135^\circ$ to radians

Multiply by $\dfrac{\pi}{180}$ :

135^\circ\left(\frac{\pi}{180}\right)

Simplify the fraction:

\frac{135\pi}{180}=\frac{3\pi}{4}

Therefore,

\boxed{135^\circ=\frac{3\pi}{4}\text{ radians}}

The degree symbol cancels during the conversion, leaving the answer in radians.

Learn by doing: Convert between degrees and radians

Click a topic below to practice the foundational skills you'll need, learn the steps, or master this skill

Practice with unlimited practice problems

Radians - Convert Degrees to Radians

Earned ?