We examine the Landau constants defined by $$G_n:=\sum_{m\,=0}^{n}\frac{1}{2^{4 m}}\,\binom{2 m}{m}^2\qquad(n=0, 1, 2, \cdots)$$ by making use of the celebrated Ramanujan formula expressing $G_n$ in terms of the Clausenian ${}_3F_2$ hypergeometric series. It is shown that it could be used to deduce other, mostly new, Ramanujan type formulas for the Landau constants involving the terminating and non-terminating hypergeometric series. In addition, by this approach we derive once again, in a simple and unified manner, almost all of the known results and also establish several new results for $G_n$. These new results include (for example) the generating function and asymptotic expansions and estimates for $G_n$.