  In a recent work [Ge {\it et al.}, arXiv: 2312. 17496 (2023)], we have
derived the polygon relation of bipartite entanglement measures that is useful
to reveal the entanglement properties of discrete, continuous, and even hybrid
multipartite quantum systems. In this work, with the information-theoretical
measures of R\'enyi and Tsallis entropies, we study the relationship between
the polygon relation and the subadditivity of entropy. In particular, the
entropy-polygon relations are derived for pure multi-qubit states and
generalized to multi-mode Gaussian states, by utilizing the known results from
the quantum marginal problem. Moreover, the equivalence between the polygon
relation and subadditivity is established, in the sense that for all discrete
or continuous multipartite states, the polygon relation holds if and only if
the underlying entropy is subadditive. As byproduct, the subadditivity of
R\'enyi and Tsallis entropies is proven for all bipartite Gaussian states.
Finally, the difference between polygon relations and monogamy relations is
clarified, and generalizations of our results are discussed. Our work provides
a better understanding of the rich structure of multipartite states, and hence
is expected to be helpful for the study of multipartite entanglement.
