Derive the relation between linear and angular velocity,or Derive $v=rw$

Consider a body rotating about an axis passing through a point $O$and perpendicular to the plane of the paper. Let $P$ be the position of a particle inside the body. If the body rotates through an angle $0$ in a time interval of $t$, the particle at $P$ reaches $P^{\prime }$ 
$\therefore$  Linear displacement $P{P}^{\prime }=r\theta$ 
$\therefore$Linear velocity, $v=\frac{P{P}^{\prime }}{t}=\frac{r\theta }{t}$
But $\frac{\theta }{t}=\omega$, the angular velocity
$\therefore$Linear velocity , $v=r\omega$