This paper provides a solution to a critical issue in large-scale Multi-User Multiple-Input Multiple-Output (MU-MIMO) communication systems: how to estimate the Signal-to-Interference-plus-Noise-Ratios (SINRs) and their expectations in MU-MIMO mode at the Base Station (BS) side when only the Channel Quality Information (CQI) in Single-User MIMO (SU-MIMO) mode and non-ideal Channel State Information (CSI) are known? A solution to this problem would be very beneficial for the BS to predict the capacity of MU-MIMO and choose the proper modulation and channel coding for MU-MIMO. To that end, this paper derives a normalized volume formula of a hyperball based on the probability density function of the canonical angle between any two points in a complex Grassmann manifold, and shows that this formula provides a solution to the aforementioned issue. It enables the capability of a BS to predict the capacity loss due to non-ideal CSI, group users in MU-MIMO mode, choose the proper modulation and channel coding, and adaptively switch between SU-MIMO and MU-MIMO modes, as well as between Conjugate Beamforming (CB) and Zero-Forcing (ZF) precoding. Numerical results are provided to verify the validity and accuracy of the solution.