Suppose that the particle P is moving in the plane, in a circle with centre at the origin O and radius r₀. Let (r₀, θ‎) be the polar coordinates of P. At an elementary level, the angular velocity may be defined to be $\dot{\theta }$, where a dot denotes differentiation with respect to time.

At a more advanced level, the angular velocity ω‎ of the particle P is the vector defined by ω‎ = θ‎˙ k, where i and j are unit vectors in the directions of the positive x- and y-axes, and k = i × j. If r and v are the position vector and velocity of P, then

where e_r = i cos θ‎ + j sin θ‎ and e_θ‎ =−i sinθ‎ + j cos θ‎ (see circular motion). By using the fact that k = e_r × e_θ‎, it follows that the velocity v is given by v = ω‎ × r.

Consider a rigid body rotating about a fixed axis, and take coordinate axes so that the z-axis is along the fixed axis. Let (r₀, θ‎) be the polar coordinates of some point of the rigid body, not on the axis, lying in the plane z =0. Then the angular velocity ω‎ of the rigid body is defined by $\mathbf{\omega }=\dot{\theta }\mathbf{k}$.

In general, for a rigid body that is rotating, such as a top spinning about a fixed point, the rigid body possesses an angular velocity ω‎ whose magnitude and direction depend on time, though the formula v = ω‎ × r still applies.