This study proposes a high-precision fast approximation method for the ℓ2-norm evaluation of 2-tuple data arrays using a rotated ℓ1-norm evaluation with fixed-point arithmetic. In several signal processing applications, such as image restoration with isotropic total variation (TV) and one with complex ℓ1-norm regularization, a large number of calculations for the 2-tuple ℓ2-norm are frequently required. To achieve a hardware (HW)-friendly calculation, the square and square root operations involved in the ℓ2-norm calculation should be adequately approximated. However, several existing techniques have been challenged with respect to approximations. Thus, in this paper, a HW-friendly approximation algorithm is proposed. The proposed method uses the fact that the upper bound of the surface of a first-order rotational cone traces a second-order cone, that is, the ℓ2-cone. As a result, less variable multiplication is required, and parallel implementation is easily achieved using fixed-point arithmetic. To demonstrate the effectiveness of the proposed method, it was applied to image restoration, and then its performance on field programmable gate arrays (FPGA) is evaluated in terms of the quality, circuit area, latency, and throughput. The effectiveness of the proposed method is verified by comparing it with typical implementations using commercial circuits.