It is known that the communication channel capacity of the additive white Gaussian noise (AWGN) channel under peak power constraint is achieved by a distribution, whose support is a finite isolated points in a scalar case and is concentric circles in a vector case. In this paper, we compare the achievable rates of phase shift keying (PSK) and quadrature amplitude modulation (QAM) input constellations for a complex-valued AWGN channel with the capacity under the peak power constraint, and compare their constellations with the capacity-achieving distributions. The comparison reveals that achievable rates of PSK and QAM are close to capacities for certain ranges of signal to noise ratio (SNR) under different forms of peak power constraint.