  Within the Skyrme-Hartree-Fock (SHF) approach, we show that for a fixed mass
number A, both the symmetry energy coefficient a_{sym}(A) in the semi-empirical
mass formula and the nuclear matter symmetry energy E_{sym}(\rho_A) at a
subsaturation reference density rho_A can be determined essentially by the
symmetry energy E_{sym}(rho_0) and its density slope L at saturation density
rho_0. Meanwhile, we find the dependence of a_{sym}(A) on E_{sym}(rho_0) or L
is approximately linear and is very similar to the corresponding linear
dependence displayed by E_{sym}(\rho_A), providing an explanation for the
relation E_{sym}(\rho_A) \approx a_{sym}(A). Our results indicate that a value
of E_{sym}(\rho_A) leads to a linear correlation between E_{sym}(rho_0) and L
and thus can put important constraints on E_{sym}(rho_0) and L. Particularly,
the values of E_{sym}(rho_0)= 30.5 +- 3 MeV and L= 52.5 +- 20 MeV are
simultaneously obtained by combining the constraints from recently extracted
E_{sym}(\rho_A=0.1 fm^{-3}) with those from recent analyses of neutron skin
thickness of Sn isotopes in the same SHF approach.
