Degree-to-Radian Conversion
TrigonometryDegree-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 $\pi$.
Example
$90$ degrees $= 90 \times (\pi/180) = \pi/2$ radians. And $\pi$ radians $= \pi \times (180/\pi) = 180$ degrees.
Key Insight
Think of it as a trade: every $180$ degrees equals $\pi$ radians. So $180^\circ = \pi$, and you can scale from there.
Definition
Because $360^\circ = 2\pi$ radians, the conversion factors are: multiply degrees by $\pi/180$ to get radians, and multiply radians by $180/\pi$ to get degrees. Common conversions: $30^\circ = \pi/6$, $45^\circ = \pi/4$, $60^\circ = \pi/3$, $90^\circ = \pi/2$, $180^\circ = \pi$, $270^\circ = 3\pi/2$, $360^\circ = 2\pi$.
Example
Convert $150^\circ$ to radians: $150 \times (\pi/180) = 150\pi/180 = 5\pi/6$. Convert $7\pi/4$ to degrees: $(7\pi/4) \times (180/\pi) = 7 \times 180/4 = 315^\circ$.
Key Insight
Scientific calculators have a mode switch (DEG/RAD). Forgetting to switch modes is one of the most common sources of error in trig calculations. When in doubt, radians are the default in advanced math.
Definition
The conversion factor $\pi/180$ arises from the proportionality of arc length to central angle: $\theta_{\text{radians}} / (2\pi) = \theta_{\text{degrees}} / 360$, giving $\theta_{\text{radians}} = \theta_{\text{degrees}} \times \pi/180$. This is a dimensionful conversion: degrees are an arbitrary unit while radians are dimensionless (arc length / radius).
Example
In complex analysis, the argument of a complex number is always given in radians: $\arg(e^{i\pi/6}) = \pi/6$, corresponding to $30^\circ$. Software like NumPy and MATLAB use radians by default in all trig functions.
Key Insight
Radian measure is invariant under circle scaling; degree measure is not. This is what makes radians "natural": they encode the pure ratio of arc to radius, independent of the circle's size, making them the correct unit for analysis on $S^1$ and for Fourier transforms.