  Radhakrishnan \emph{et.al} [Phys. Rev. Lett. \textbf{124}, 110401 (2020)]
proposed quantum discord to multipartite systems and derived explicit formulae
for any states. These results are significant in capturing quantum correlations
for multi-qubit systems. In this paper, we propose the geometric measure of
multipartite quantum discord and obtain results for a large family of
multi-qubit states. Intriguingly, our results demonstrate that for the family
of states, multipartite geometric discord decreases exponentially as the number
of parties increases while multipartite quantum discord not, therefore, we
conclude that multipartite geometric discord could not reflect multipartite
quantum discord in all cases. Furthermore, we investigated the dynamic behavior
of geometric discord for the family of two-, three- and four-qubit states under
phase noise acting on the first qubit. And we discover that sudden change of
multipartite geometric discord can appear when phase noise act only on one part
of the two-, three- and four-qubit states.
