Degree-to-Radian Conversion
Degree-to-radian conversion is the process of changing an angle measurement from degrees to radians using the ratio $\pi/180$.
Formula
\text{radians} = \text{degrees} \times \frac{\pi}{180}; \quad \text{degrees} = \text{radians} \times \frac{180}{\pi}
Definition
To convert degrees to radians, multiply by $\pi$ and divide by $180$; to go the other way, multiply by $180$ and divide by $\pi$. Because $360^\circ = 2\pi$ radians, the conversion factors are $\text{radians} = \text{degrees} \times \pi/180$ and $\text{degrees} = \text{radians} \times 180/\pi$, arising from the proportionality of arc length to central angle, $\theta_{\text{radians}} / (2\pi) = \theta_{\text{degrees}} / 360$. This is a dimensionful conversion: degrees are an arbitrary unit while radians are dimensionless (arc length divided by radius).
Example
$90$ degrees $= 90 \times (\pi/180) = \pi/2$ radians, and $\pi$ radians $= 180$ degrees; common conversions include $30^\circ = \pi/6$, $45^\circ = \pi/4$, $60^\circ = \pi/3$, $150^\circ = 5\pi/6$, and $270^\circ = 3\pi/2$. Converting $7\pi/4$ to degrees gives $(7\pi/4) \times (180/\pi) = 315^\circ$. In complex analysis, the argument of a complex number is always given in radians, $\arg(e^{i\pi/6}) = \pi/6$ (corresponding to $30^\circ$), and software like NumPy and MATLAB use radians by default in all trig functions.
Key Insight
Think of it as a trade: every $180$ degrees equals $\pi$ radians, and you can scale from there. Scientific calculators have a DEG/RAD mode switch, and forgetting to switch is one of the most common sources of error in trig calculations; radians are the default in advanced math. Radian measure is invariant under circle scaling while degree measure is not, which is what makes radians "natural": they encode the pure ratio of arc to radius, independent of the circle's size, the correct unit for analysis on $S^1$ and for Fourier transforms.