An important unit in describing rotational motion is the radian, abbreviated as rad. For its derivation, a circular segment of length $s$. This distance can be calculated from the circumference of a circle $U=2\pi r$ by multiplying it by the ratio of the included angle $\alpha$ in angular units (abbreviated deg) and the angle of the complete circle:
\begin{equation}
s = \underbrace{2\pi \frac{\alpha[\mathrm{deg}]}{360^\circ}}_{\alpha[\mathrm{rad}]}r
\end{equation}
The term before the radius, which contains the angle $\alpha$ in degrees, can be summarized as the same angle in radians, as shown. The usefulness of this unit is evident in the relationship between the length of a circle's arc and the radius when the angle is given directly in radians. Then you can simply write:
\begin{equation}
s = \alpha r
\end{equation}
This gives the conversion factor between angle and radian measure:
\begin{equation}
{\alpha[\mathrm{rad}] = \pi\frac{\alpha[\mathrm{deg}]}{180^\circ}}
\end{equation}
The full angle $360^\circ$ in degrees therefore corresponds to the quantity $2\pi$ in radians.
If a body moves along a curved path, for example, a circle, then the speed along this path is called \textit{orbital velocity}\index{orbital velocity} and is usually denoted by $\vec{v}$. The definition of orbital velocity is then given as:
\begin{equation}
\vec{v} = \frac{\mathrm{d}\vec{r}}{\mathrm{d}t}
\end{equation}
In the case of a circular orbit, this can be easily calculated from the circumference of the circle and the time elapsed during its orbit. The time $T$ that the body requires for one orbit is called the \textit{orbital period}\index{orbital period}. The orbital velocity of the body is thus given by
\begin{equation}
v = \frac{\Delta s}{\Delta t} = \frac{2\pi r}{T}
\end{equation}
The term $2\pi/T$ indicates the time required to cover the total angle $2\pi$ and is therefore called angular velocity. Together with the definition of radians, the following relationship between orbital and angular velocity results:
\begin{equation}\label{eq:angular velocity}
{v = \omega r}
\end{equation}
It should be noted that the angular velocity must be specified in radians or first converted to this unit.
In spatial motion, it is also important to note that the angular velocity, like the orbital velocity, is a vector defined to be perpendicular to the radius and velocity vectors.