A digit-recurrence algorithm for computing the Euclidean norm of a 3-dimensional vector is proposed. Starting from the vector component with the highest order of magnitude as the initial value of partial result, correcting-digits produced by the recurrence are added to it step by step. Partial products of the squares of the other two components are added to the residual, step by step. The addition/subtractions in the recurrence are performed without carry/borrow propagation by the use of a redundant representation of the residual. An extension of the on-the-fly conversion algorithm is used for updating the partial result. Different specific versions of the algorithm are possible, depending on the radix, the redundancy factor of the correcting-digit set, the type of representation of the residual, and the correcting-digit selection function.