How many sides does a regular polygon have if each of its interior angles is $165^{0}$ ?

Number of sides

= 24

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Number of sides

= 30

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Number of sides

= 36

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Number of sides

= 42

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Solution

The correct option is A Number of sides $= 24$
$- L e t t h e g i v e n p o l y g o n h a v e n s i d e s . ∴ t h e s u m o f t h e i n t e r i o r a n g l e s = (n - 2) 180^{o} & t h e n u m b e r o f a n g l e s i s n . A g a i n t h e p o l y g o n i s a r e g u l a r o n e . S o t h e i n t e r i o r a n g l e s a r e e q u a l t o e a c h o t h e r . L e t e a c h i n t e r i o r a n g l e b e θ .$
$∴ θ = \frac{(n - 2) 180^{o}}{n} ⟹ n = \frac{360^{o}}{180^{o} - θ} .$
$G i v e n t h a t θ = 165^{o}$
$∴ n = \frac{360^{o}}{180^{o} - 165^{o}} = 24$
$A n s - 24 s i d e s .$

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