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?
A full circle is $360^\circ$ and also $2\pi$ radians. Thus $360^\circ = 2\pi\text{ rad}$. We simplify by dividing by two.
Since circumference =2pier if we take half circle it would be pier in which rotation of angle is 180 degree and  arc length would be pie*r then we get theta =pie =180 degree