Let N_k(n,r,a) denote the number of incongruent solutions of the quadratic congruence a₁x₁² + ··· + a_kx_k² ≡ n (mod r), where a = (a₁, ... ,a_k) ∈ Z_k, n ∈ Z, r ∈ N. We give short direct proofs for certain less known compact formulas on N_k(n,r,a), valid for r odd, which go back to the work of Minkowski, Bachmann and Cohen. We also deduce some other related identities and asymptotic formulas which do not seem to appear in the literature.