Question

The ratio of an interior angle to an exterior angle of a regular polygon is 5 : 2. What is the number of sides of the polygon ?​

Answer 1

Answer:

$\huge\underline\bold {Answer:}$

Interior angle of polygon of n sides = (n – 2) × 180°/n

Exterior angle of a polygon of n sides = 360°/n

Given, (n – 2) × 180°/n : 360°/n = 5 : 2

=> (n – 2) × 180°/n × 360°/n = 5/2

=> (n – 2)/2 = 5/2

=> 2(n – 2) = 10

=> 2n – 4 = 10

=> 2n = 14

=> n = 7.

Therefore number of sides of the polygon is 7.

Answer 2

$\bigstar \mid$QUESTION :

• The ratio of an interior angle to an exterior angle of a regular polygon is 5 : 2. What is the number of sides of the polygon ?

$\bigstar \mid$GIVEN :

• Ratio of an interior and exterior angle of a regular polygon is 5:2 respectively.

$\bigstar \mid$TO FIND :

• The no. of sides of the polygon.

$\bigstar \mid$ SOLUTION :

We know that,

• Interior angle of a polygon having n sides :-

$\sf = \frac{(n - 2 )\times 180\degree}{ n} \\ \sf$

And,

• Exterior angle of a polygon having n sides :-

$\sf \: = \frac{360 \degree}{n}$

Now,

According to the question,

We can say that,

$\implies \sf \: \frac{(n - 2) \times 180 \degree}{n} \ratio \: \frac{360 \degree}{n} = 5 \ratio2$

$\implies \sf \frac{(n - 2) \times \cancel{180} \degree}{ \cancel{n}} \times \frac{ \cancel{n}}{ \cancel{360} \degree} = 5 \ratio2$

$\sf \implies \sf \: \frac{(n - 2)}{2} = \frac{5}{2}$

$\sf \implies \: 2(n - 2) = 5 \times 2$

$\sf \implies \: 2n - 4 = 10$

$\sf \implies \: 2n = 10 + 4$

$\sf \implies \: 2n = 14$

$\sf \implies \: n = \frac{ \cancel{14}}{ \cancel2}$

${ \underline{ \boxed{\bf \therefore \: n = 7}}}$

Hence,

• No. of sides of polygon is 7.

CHECK POINT :

Putting the value of n :

• In the formula of Interior angle of a polygon having n sides :-

$\sf = \frac{(n - 2 )\times 180\degree}{ n} \\ \sf$

And,

• Exterior angle of a polygon having n sides :-

$\sf \: = \frac{360 \degree}{n}$

$\sf \implies \: \frac{(n - 2 )\times 180\degree}{ n} \ratio \frac{360}{7} =5 \ratio2$

$\sf \implies \: \frac{(7 - 2) \times 180 \degree}{7} \ratio \frac{360}{7} = 5 \ratio2$

$\sf \implies \frac{ 5\times \cancel{180}}{ \cancel7} \times \frac{ \cancel7}{ \cancel{360}} = 5 \ratio2$

$\sf \implies \: \frac{5}{2} = 5 \ratio2$

${\underline {\boxed{ \bf \therefore \: \: \: 5 \ratio2 = 5 \ratio2}}}$

L. H. S = R. H. S.

$\sf { \underline{ \green{ Hence,\purple V \red e \orange r \pink i \blue f \green i \red e \pink d : -}}}$