An approximate theory, including effects of the density, for the equilibrium properties of a polymer lattice model (bond-fluctuation model) is presented and compared with Monte Carlo simulations. Effects of the density are taken into account by balancing the volume requirements of the individual bond vectors with the pressure that a melt exerts at a given density. This simple theory yields rather accurate (to within a few percent) estimates for the chain length and temperature dependence of various quantities that scan different length scales of the polymers (bond length, radius of gyration, etc.). With the inclusion of the density, one can also design an improved definition of an internal temperature of the melt, which is distinct from the external temperature of the surrounding heat bath. When the melt falls out of equilibrium during the vitrification process, this internal temperature is expected to measure the temperature that the melt actually experiences, and may thus serve to eliminate the nonequilibrium effects from the simulation data. The extent to which this expectation is borne out by a comparison with the simulation is addressed.