If $A_1,A_2,\mathrm{........},A_n$ are the vertices of a regular plane polygon with n sides and O is the centre then $\sum _{i=1}^{n-1}\left(\overline{O{A}_i}\times \overline{O{A}_{i+1}}\right)=$

$$\begin{gathered} Now \overline{O{A}_1}\times \overline{O{A}_2} =\left|\overline{O{A}_1}\right|\left|\overline{O{A}_2} \right| \sin \theta n^{^} \\ \overline{ O{A}_1},\overline{O{A}_2},.........\overline{O{A}_n } will be of same length and subtend same angle at the centre \\ \therefore \underset{ }{\overset{ }{ \sum _{i=1}^{n-1} (O{A}_i \times O{A}_{i+1)}}}=(n-1)\left|\overline{O{A}_1}\right|\left|\overline{O{A}_2} \right| \sin \theta n^{^} \\ =(n-1)\overline{(O{A}_1}\times \overline{O{A}_2}) \end{gathered}$$