Relationship between Degree and Radian Measures

Degrees and radians are two different units for measuring the same underlying quantity — an angle — much as Celsius and Fahrenheit are two units for temperature. Radian measure is the more natural unit mathematically: it is defined directly from geometry (an arc-length-to-radius ratio) rather than from an arbitrary historical choice like $360$, and — as the sector-area comparison below shows — calculations come out noticeably simpler in radians.

The conversion relation. On a unit circle, one full rotation corresponds to $360^\circ$; the same full rotation, traced once around the circle, covers a distance equal to the circumference, $2\pi$. So one full rotation is also $2\pi$ radians. Equating the two descriptions of the same rotation:

Both conversion directions follow immediately:

A standard table worth knowing:

Also, $1$ radian $\approx 57^\circ17'45''$ and $1^\circ \approx 0.017453$ radians — useful for sanity-checking that a converted answer is roughly the right size.

Why radians simplify calculation: the sector-area example. For a sector of a circle of radius $r$ with central angle $\theta$:

$\text{Area of sector} = \begin{cases} \left(\dfrac{\pi r^2}{360^\circ}\right)\theta & \theta \text{ in degree measure} \\[2mm] \left(\dfrac{\pi r^2}{2\pi}\right)\theta = \dfrac{r^2\theta}{2} & \theta \text{ in radian measure.} \end{cases}$ …