The exterior and interior angles of a regular polygon are in the ratio 2:3. Find the measure of each interior and exterior angle. Also find the number of sides of a regular polygon.

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Solution

Dear Student,

$L e t n b e r t h e n u m b e r o f s i d e s o f t h e r e g u l a r p o l y g o n . W e k n o w, E a c h e x t e r i o r a n g l e o f t h e r e g u l a r p o l y g o n = \frac{360 °}{n} a n d e a c h i n t e r i o r a n g l e o f t h e r e g u l a r p o l y g o n = 180 ° - \frac{360 °}{n} A c c o r d i n g t o t h e q u e s t i o n : \frac{\frac{360 °}{n}}{180 ° - \frac{360 °}{n}} = \frac{2}{3} \Rightarrow \frac{\frac{360 °}{n}}{\frac{180 ° n - 360 °}{n}} = \frac{2}{3} \Rightarrow \frac{360 °}{180 ° n - 360 °} = \frac{2}{3} \Rightarrow \frac{180 ° \times 2}{180 ° (n - 2)} = \frac{2}{3} \Rightarrow \frac{2}{n - 2} = \frac{2}{3} \Rightarrow 2 \times 3 = 2 (n - 2) \Rightarrow n - 2 = 3 \Rightarrow n = 3 + 2 \Rightarrow n = 5 S o, t h e n u m b e r o f s i d e s o f t h e r e g u l a r p o l y g o n = n = 5 M e a s u r e o f e a c h e x t e r i o r a n g l e = \frac{360 °}{n} = \frac{360 °}{5} = 72 ° M e a s u r e o f e a c h i n t e r i o r a n g l e = 180 ° - \frac{360 °}{n} = 180 ° - \frac{360 °}{5} = 180 ° - 72 ° = 108 °$

Regards

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Similar questions

The measure of each interior angle of a regular polygon is five times the measure of its exterior angle. Find :

(i) measure of each interior angle :

(ii) measure of each exterior angle and

(iii) number of sides in the polygon.

2 : 7

2 : 3

2 : 1

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