Why do we use $\frac{\pi}{180}$ to convert from degrees to radians? If I want to convert from degrees to radians, I can use the function that takes degree value as an input, multiplies it with $\frac{\pi}{180}$ and returns the radian value: $\operatorname{DtoR}(d)=d \times \frac{\pi}{180}$.
And if I want to go from radians to degrees I need to only go backwards and  divide radian value with $\frac{\pi}{180}$ (e.g. multiply it with $\frac{180}{\pi}$): $\operatorname{RtoD}(r)=r \times \frac{180}{\pi}$.
My question is this: Why does multiplying/dividing with $\frac{\pi}{180}$ converts degrees into radians/radians into degrees? Why exactly that number, not some other? Also, does this work only for unit circle, or for any circle?

The radian is defined as the plane angle subtended by any circular arc divided by its radius.
When the circular arc is actually congruent to the circle, the length is $2\pi r=2\pi=$ $\tau$ (for a unit circle). The angle subtended by this arc is $360^\text{o}$, and therefore $1\:\text{radian}=\frac{360}{\tau}=\frac{180}{\pi}$.
So: $$r\text{ radians}=\text{d}\cdot \frac{180}{\pi}\\ d\text{ degrees} = \frac{r}{\frac{180}{\pi}}=\frac{180r}{\pi}$$
