The Lagrangian density of an r-uniform hypergraph F is r! multiplying the supremum of the Lagrangians of all F-free r-uniform hypergraphs. For an r-graph H with t vertices, it is clear that πλ(H)≥r!λ(Kt−1r). We say that an r-uniform hypergraph H with t vertices is λ-perfect if πλ(H)=r!λ(Kt−1r). A theorem of Motzkin–Straus implies that all 2-uniform graphs are λ-perfect. It is interesting to explore what kind of hypergraphs are λ-perfect. A hypergraph is linear if any 2 edges have at most 1 vertex in common. We propose the following conjecture: (1) For r≥3, there exists n such that a linear r-uniform hypergraph with at least n vertices is λ-perfect. (2) For r≥3, there exists n such that if G,H are λ-perfect r-graphs with at least n vertices, then G⨆H is λ-perfect. Regarding this conjecture, we obtain a partial result: Let S2,t={123,124,125,126,…,12(t+2)}. (An earlier result of Sidorenko states that S2,t is λ-perfect (Sidorenko, 1989).) Let H be a λ-perfect 3-graph with s vertices. Then F=S2,t⨆H is λ-perfect if s≥3 and t≥3.There was no known result on Lagrangian densities of hypergraph cycles and there were 3 unsolved cases for 3-uniform graphs spanned by 3 edges: a linear cycle of length 3: C33={123,345,561}, the generalized triangle: F5={123,124,345} and K43−={123,124,134}. In this paper, we obtain the Lagrangian density of F5 which is an example of non-λ-perfect 3-uniform graph. We show that C33 is λ-perfect, and among all C33-free 3-graphs G, only those hypergraphs containing K53 achieve the Lagrangian λ(K53). The Turán densities of extensions of the above hypergraphs can be obtained by applying a transference technique of Pikhurko.