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 acceleration may be defined to be $\ddot{\theta }$, where a dot denotes differentiation with respect to time.

At a more advanced level, the angular acceleration α‎ of the particle P is the vector defined by α‎ = $\dot{\mathbf{\text{ω}}}$, where ω‎ is the angular velocity. Let i and j be unit vectors in the directions of the positive x- and y-axes and let k = i × j. Then, in the case above of a particle moving along a circular path, ω‎ = $\dot{\theta }$ k and α‎ = $\ddot{\theta }$ k. If r, v and a are the position vector, velocity and acceleration of P, then

where e_r = i cosθ‎ + + j sinθ‎ and e_θ‎=−i sinθ‎ + j cosθ‎ (see circular motion). Using the fact that v = ω‎ × r, it follows that the acceleration a = α‎ × r + ω‎ × (ω‎ × r).