Converting Between Degrees and Radians

Degrees and radians are both units for measuring angles, but they are built on different ideas. A degree splits a full circle into 360 equal pieces, a system that goes back to ancient astronomy. A radian, on the other hand, is based on the circle's own radius: one radian is the angle created when the arc length around a circle equals the radius of that circle. Because a full circle's circumference is \(2\pi r\), going all the way around corresponds to an angle of \(2\pi\) radians — and that same angle is 360 degrees.

Since both units describe the same rotation, you can always convert from one to the other. This skill shows up constantly once you start working with the trigonometric ratios of angles in radians, since many trig problems switch between the two units depending on the context.

Because 360 degrees equals \(2\pi\) radians, dividing both sides by 2 gives the key relationship:

\(180^\circ = \pi \) radians\(\)

From this single fact, both conversion formulas fall out directly.

Degrees to radians: multiply the number of degrees by \(\frac{\pi}{180}\).

\(\)radians\( = \)degrees\( \times \frac{\pi}{180}\)

Radians to degrees: multiply the number of radians by \(\frac{180}{\pi}\).

\(\)degrees\( = \)radians\( \times \frac{180}{\pi}\)

Notice that these two formulas are reciprocals of each other, so you never need to memorize both separately — just remember that \(180^\circ = \pi\) radians and flip the fraction depending on which direction you're converting.

Convert \(150^\circ\) to radians.

\(150 \times \frac{\pi}{180} = \frac{150\pi}{180} = \frac{5\pi}{6}\)

So \(150^\circ = \frac{5\pi}{6}\) radians. Always simplify the fraction at the end, since most angles convert to "nice" fractions of \(\pi\).

Convert \(\frac{7\pi}{4}\) radians to degrees.

\(\frac{7\pi}{4} \times \frac{180}{\pi} = \frac{7 \times 180}{4} = \frac{1260}{4} = 315\)

So \(\frac{7\pi}{4}\) radians equals \(315^\circ\). Notice that the \(\pi\) terms cancel out, which always happens when converting radians (given as a multiple of \(\pi\)) back to degrees.

Certain angles appear so often in trigonometry that it helps to know their radian equivalents by heart. These are the same angles you will use when applying the ASTC rule or finding a reference angle for a given angle.

Degrees	Radians
0 degrees	0
30 degrees	\(\frac{\pi}{6}\)
45 degrees	\(\frac{\pi}{4}\)
60 degrees	\(\frac{\pi}{3}\)
90 degrees	\(\frac{\pi}{2}\)
180 degrees	\(\pi\)
270 degrees	\(\frac{3\pi}{2}\)
360 degrees	\(2\pi\)

The most frequent error is multiplying by the wrong fraction, for example using \(\frac{180}{\pi}\) when converting degrees to radians instead of \(\frac{\pi}{180}\). A quick sanity check helps: radian measures of angles smaller than a full circle are usually small numbers (often less than about 6.3, since \(2\pi \approx 6.28\)), while degree measures for the same angles are much larger numbers, up to 360. If your converted value looks the wrong size, check which fraction you used.

Another common slip is forgetting to simplify the fraction of \(\pi\) at the end, or forgetting the \(\pi\) symbol entirely when writing a radian answer. Radian measures should generally be left in exact form (like \(\frac{5\pi}{6}\)) rather than converted to a decimal unless a problem specifically asks for a decimal approximation.

Once you're comfortable converting between degrees and radians, the next step is usually applying that skill directly on the unit circle and in formulas like the law of cosines, where angles can be given in either unit. Practice converting a mix of degree and radian values until the two formulas feel automatic.

Related lessons

• Trigonometry Help