Results are reported on the average radius of gyration <S2N(L)≳L for self-avoiding walks of size N steps having span L generated on the simple cubic lattice. Limiting behavior was fitted to expressions of the form <S2N(L)≳L∼ANγ(L) where γ(L) was found to vary from γ(O)∼1.50 to γ(N)∼1.20. In addition, the limiting ratio <S2N(L)≳L/<R2N(L)≳L was investigated. It was found that <S2N(O)≳L/<R2N(O)≳L →0.136, whereas <S2N(N)≳L/<R2N(N)≳L →0.155. In the interval O⩽L⩽N, γ(L)→1.0, and <S2N(L)≳L/<R2N(L)≳L →0.167, indicating a conformational change in the chains to a more compact randomlike configuration.