  Question

Find the number of sides of a regular polygon when each of its angles has a measures of (i) 160∘ (ii) 135∘ (iii) 175∘ (iv) 162∘ (v) 150∘.

Solution

In a n-sided regular polygon, each angle = 2n−4n right angles or [2n−4n×90∘] (i) When each interior angle = 160∘ Let number of sides of regular polygon=n  ∴2n−4n×90∘=160∘2n−4n=160∘90∘=169 By cross multiplication: 18n−36=16n⇒18n−16n=36⇒2n=36⇒n=362=18 ∴ Regular polygon is of 18 sided.   (ii) Each interior angle = 135∘ Let number of sides of regular polygon = n 2n−4n×90∘=135∘⇒2n−4n=13590=32 By cross multiplication: 4n−8=3n⇒4n−3n=8⇒n=8 ∴ The regular polygon has 8 sides. (iii) Each interior angle = 175∘ Let number of sides of regular polygon = n  ∴2n−4n×90∘=175∘⇒2n−4n=17590=3518 By cross multiplication: 36n-72=35n ⇒36n−35n=72⇒n=72 ∴ Regular polygon is of 72 sides. (iv) Each interior angle = 162∘ Let number of sides of regular polygon=n  ∴2n−4n×90∘=162∘ ⇒2n−4n=16290=95 By cross multiplication: 10n-20=9n ⇒10n−9n=20⇒n=20 ∴ Regular polygon is 20sides. (v) Each interior angle = 150∘ Let number of sides of regular polygon = n ∴2n−4n×90∘=150∘⇒2n−4n=15090=53 By cross multiplication, 6n−12=5n⇒6n−5n=12⇒n=12 ∴ The regular polygon of 12 sides.    MathematicsRD SharmaStandard VIII

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