Question

The exterior angle of a regular polygon is one-third of its interior angle. How many sides has the Polygon ?​

Answer 1

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$\bf\underline{\underline{\orange{Question:-}}}$

The exterior angle of a regular polygon is one-third of its interior angle. How many sides has the Polygon ?

$\bf\underline{\underline{\orange{AnSwEr:-}}}$

Let number of sides = n.

Given that:-

exterior angle = 1/3 of interior angle.

or, 360°/n = 1/3.( 180° -360°/n)

or, 360°/n = 60° - 120°/n.

or, 360°/n. + 120°/n = 60°

or, 480°/n = 60°.

or, n = 480°/60° = 8 sides

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Answer 2

Question :-

• The exterior angle of a regular polygon is one-third of its interior angle. How many sides has the Polygon ?

To Find :-

• Sides of the polygon.

Solution :-

Let there be n sides of the polygon. Then,

• Each Exterior Angle = $\bigg( \frac{360}{n} \bigg)$

• Each Interior Angle = $\bigg( \frac{2n \: - \: 4}{n} \: \times \: 90 \bigg)$

$\begin{gathered}\begin{gathered}\underline{\boldsymbol{According\: to \:the\: Question :}} \\\end{gathered}\end{gathered}$

$\qquad \quad {:} \longrightarrow \sf \boxed{\bf{Exterior \: Angle \: = \: \frac{1}{3}\: \times \: ( Interior \: Angle ) }}$

$\qquad \quad {:} \longrightarrow \sf {\sf{ \frac{360}{n} \: = \: \frac{1}{3}\: \times \: \frac{2n \: - \: 4}{n} \: \times \: 90 }}$

$\qquad \quad {:} \longrightarrow \sf {\sf{ \frac{360}{n} \: = \: \frac{1}{ \cancel3}\: \times \: \cancel{90} \: \times \: \frac{2n \: - \: 4}{n} \: }}$

$\qquad \quad {:} \longrightarrow \sf {\sf{ \frac{360}{n} \: = \: 60 \: \times \: \frac{n \: - \: 2}{n} \: }}$

$\qquad \quad {:} \longrightarrow \sf {\sf{ \frac{360}{60} \: = \: n\: \times \: \frac{n \: - \: 2}{n} \: }}$

$\qquad \quad {:} \longrightarrow \sf {\sf{ \cancel \frac{360}{60} \: = \: \cancel {n}\: \times \: \frac{n \: - \: 2}{ \cancel{n}} \: }}$

$\qquad \quad {:} \longrightarrow \sf {\sf{ 6 \: = \: n \: - \: 2 \: }}$

$\qquad \quad {:} \longrightarrow \sf {\sf{ 6 \: + \: 2 \: = \: n \: }}$

$\qquad \quad {:} \longrightarrow \sf {\bf{8 \: = \: n \: }}$

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Thus, The polygon has 8 sides

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