What if the radian measure does not contain $\pi$?

Multiply by $\frac{180^\circ}{\pi}$ as usual and evaluate numerically. For example, $2 ext{ radians} = 2 imes \frac{180^\circ}{\pi} \approx 114.59^\circ$.

Are negative angles converted differently?

No. The conversion factor is a positive scalar. A negative angle in degrees yields a negative angle in radians, and vice versa.

Do I need to write "radians" as a unit in the final answer?

Radian is a dimensionless unit. It is often omitted in calculus and advanced trigonometry, but writing it is standard practice when explicitly converting units to avoid ambiguity.

How to convert between degrees and radians

Converting between degrees and radians requires multiplying the angle by a conversion factor derived from the relationship $\pi ext{ radians} = 180^\circ$ . Use this method whenever a formula requires a specific angular unit, such as finding arc length or evaluating trigonometric derivatives.

The setup

Identify the starting unit of your angle. If the angle is in degrees, the target unit is radians, requiring the conversion factor $\frac{\pi}{180^\circ}$ . If the angle is in radians, the target unit is degrees, requiring the conversion factor $\frac{180^\circ}{\pi}$ .

The steps

Write the initial angle measurement with its specified unit.
Multiply the angle by the correct conversion factor ( $\frac{\pi}{180^\circ}$ for degrees to radians; $\frac{180^\circ}{\pi}$ for radians to degrees).
Cancel the original units algebraically.
Simplify the resulting fraction.

Checking the result

Confirm that typical reference angles match standard known values (e.g., $90^\circ$ should result in $\frac{\pi}{2}$ radians). Verify that the units cancelled properly; an answer in degrees must retain the degree symbol ( $^\circ$ ).

Common errors

The most frequent error is inverting the conversion factor. Multiplying a degree measure by $\frac{180^\circ}{\pi}$ yields squared degrees per radian, which is dimensionally incorrect. Always align the denominator's unit with the starting angle's unit.

Worked example

Convert $210^\circ$ to radians, and convert $\frac{5\pi}{4}$ radians to degrees.

Part 1: Convert $210^\circ$ to radians. $210^\circ imes \frac{\pi}{180^\circ}$ $= \frac{210\pi}{180}$ $= \frac{7\pi}{6} ext{ radians}$

Part 2: Convert $\frac{5\pi}{4}$ radians to degrees. $\frac{5\pi}{4} imes \frac{180^\circ}{\pi}$ $= \frac{5 imes 180^\circ}{4}$ $= 5 imes 45^\circ$ $= 225^\circ$

FAQ

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Answer form

References: OpenStax Algebra and Trigonometry, Chapter 7: The Unit Circle · Stewart Calculus, Appendix D: Trigonometry