Starting from the graphene layer, the bandgap engineering of carbon nanotubes (CNTs) and graphene nanoribbons (GNRs) is described by applying an appropriate boundary condition. Linear E-k relationship of graphene transforms to a parabolic one as momentum vector in the tube direction is reduced to dimensions smaller than inverse of the tube diameter of a CNT. Similar transition is noticeable for narrow width of a GNR. In this regime, effective mass and bandgap expressions are obtained. A CNT or GNR displays a distinctly 1D character suitable for applications in quantum transport.