Arc length $l$ can be calculated either by degrees or radians:
$l = 2\pi r \times \dfrac{\theta}{360} \text{ (in degrees)}$
$l = r\theta \text{ (in radians)}$
We can multiply the degree equation by $\dfrac{180}{\pi}$ and obtain the radian equation.
However, when we convert a $\theta$ in degrees to radians, we multiply it by $\dfrac{\pi}{180}$ instead. $\dfrac{180}{\pi}$ is used to convert radians to degrees.
I'm sure there's some reasoning behind but I can't seem to figure it out...
$\endgroup$ 32 Answers
$\begingroup$We know that a measure of $2\pi$ radians corresponds to $360$ degree, so , if $\theta_r$ is the measure in radiant and $\theta °$ is the measure in degree of the same angle we have the proportion: $$ \theta_r : \theta°=2\pi : 360° $$ (note that this is true because the two scales are proportional) from which we have: $$ \theta_r=\theta ° \frac{2 \pi}{360} $$
and the length of the arc is $$ l= r \theta_r= r \theta ° \frac{2 \pi}{360} $$
$\endgroup$ 1 $\begingroup$We know from geometry that the circumference $C$ of a circle with radius $r$ is
$$C=2\pi r.$$
Definition. Say a circle is located at the origin. Look at an arc whose length is exactly one radius. The angle of this sector with the positive x-axis is then exacly one radian. See here for an animation.
So if the arc length is $r$, the angle is $1 \ \mathrm{rad}.$ The angle of the entire circle is thus $2\pi$ times one radian.
$$\theta_{\text{entire circle}}=360^\circ=2\pi \ \mathrm{rad}.$$
Dividing by two,
$$\tag{*}\theta_{\text{half-circle}}=180^\circ=\pi \ \mathrm{rad}. $$
One degree is therefore
$$1^\circ=\frac{\pi}{180}\ \mathrm{rad}.$$
From $(*)$ you can derive the opposite relationship as well. Note also that usually the $\mathrm{rad}$ is put into parentheses or dropped altogether.
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