Question

what is the number of sides does a regular polygon have if each of interior angle is 165

Answer 1

Answer :

We know formula of each angle of regular polygon :

Measure of each interior Angle = ${\left[\frac{180\left(\mathrm{n}-2\right)}{\mathrm{n}}\right]}^{°}$ , Here n = Number of sides of regular polygon , So

$$\begin{gathered} {\left[\frac{180\left(\mathrm{n}-2\right)}{\mathrm{n}}\right]}^{°}=165° \\ \Rightarrow {\left[\frac{\left(\mathrm{n}-2\right)}{\mathrm{n}}\right]}^{°}=\frac{165°}{180°} \\ \Rightarrow \left[\frac{\mathrm{n}}{\mathrm{n}}-\frac{2}{\mathrm{n}}\right]=\frac{11}{12} \\ \Rightarrow 1-\frac{2}{\mathrm{n}}=\frac{11}{12} \\ \Rightarrow 1-\frac{11}{12}=\frac{2}{\mathrm{n}} \\ \Rightarrow \frac{12-11}{12}=\frac{2}{\mathrm{n}} \\ \Rightarrow \frac{1}{12}=\frac{2}{\mathrm{n}} \\ \Rightarrow \mathrm{n}=2\times 12 \\ \mathbf{\Rightarrow }\mathbf{}\mathbf{n}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{}\mathbf{}\mathbf{24} \end{gathered}$$

So,

Number of sides of given polygon = 24 ( Ans )