Question

11. calculate the number of sides of a regular polygon if
(ii) the ratio of an exterior angle to an interior angle is 2:7

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{the}\mathrm{number}\mathrm{of}\mathrm{sides}\mathrm{of}\mathrm{regular}\mathrm{polygon}=\mathrm{n} \\ \mathrm{Measure}\mathrm{of}\mathrm{each}\mathrm{interior}\mathrm{angle}=\frac{\left(\mathrm{n}-2\right)\times 180°}{\mathrm{n}} \\ \mathrm{Measure}\mathrm{of}\mathrm{each}\mathrm{exterior}\mathrm{angle}=\frac{360°}{\mathrm{n}} \\ \mathrm{Now},\mathrm{according}\mathrm{to}\mathrm{question}: \\ \frac{{\displaystyle \frac{360°}{\mathrm{n}}}}{{\displaystyle \frac{\left(\mathrm{n}-2\right)\times 180°}{\mathrm{n}}}}=\frac{2}{7} \\ \Rightarrow \frac{360°}{\left(\mathrm{n}-2\right)\times 180°}=\frac{2}{7} \\ \Rightarrow \frac{2}{\mathrm{n}-2}=\frac{2}{7} \\ \Rightarrow \mathrm{n}-2=7 \\ \Rightarrow \mathrm{n}=7+2 \\ \Rightarrow \mathrm{n}=9 \\ \mathrm{So},\mathrm{number}\mathrm{of}\mathrm{sides}\mathrm{of}\mathrm{regular}\mathrm{polygon}=9 \end{gathered}$$