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

$\sf {\underline{\bf \red {QUESTION :- }}}$

• 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 ?

$\sf {\underline{\bf \orange{GIVEN :- }}}$

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

$\sf {\underline{\bf \blue{TO \: FIND :- }}}$

• The no. of sides of the polygon.

$\sf {\underline{\bf \green{SOLUTION :- }}}$

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,

$\to \sf \: \frac{(n - 2) \times 180 \degree}{n} \ratio \: \frac{360 \degree}{n} = 5 \ratio2 </strong><strong>[</strong><strong>/</strong><strong>tex]</strong></p><p></p><p></p><p><strong>[</strong><strong>tex</strong><strong>]</strong><strong> \implies \sf \frac{(n - 2) \times \cancel{180} \degree}{ \cancel{n}} \times \frac{ \cancel{n}}{ \cancel{360} \degree} = 5 \ratio2 </strong><strong>[</strong><strong>/</strong><strong>tex</strong><strong>]</strong></p><p></p><p></p><p><strong>[</strong><strong>tex</strong><strong>]</strong><strong> \sf \implies \sf \: \frac{(n - 2)}{2} = \frac{5}{2} </strong><strong>[</strong><strong>/</strong><strong>tex</strong><strong>]</strong></p><p></p><p></p><p><strong>[</strong><strong>tex</strong><strong>]</strong><strong> \sf \implies \: 2(n - 2) = 5 \times 2 </strong><strong>[</strong><strong>/</strong><strong>tex</strong><strong>]</strong></p><p></p><p></p><p><strong>[</strong><strong>tex</strong><strong>]</strong><strong> \sf \implies \: 2n - 4 = 10 </strong><strong>[</strong><strong>/</strong><strong>tex</strong><strong>]</strong></p><p></p><p></p><p><strong>[</strong><strong>tex</strong><strong>]</strong><strong> \sf \implies \: 2n = 10 + 4 </strong><strong>[</strong><strong>/</strong><strong>tex</strong><strong>]</strong></p><p></p><p></p><p><strong>[</strong><strong>tex</strong><strong>]</strong><strong> \sf \implies \: 2n = 14 </strong><strong>[</strong><strong>/</strong><strong>tex</strong><strong>]</strong></p><p></p><p></p><p><strong>[</strong><strong>tex</strong><strong>]</strong><strong> \sf \implies \: n = \frac{ \cancel{14}}{ \cancel2} </strong><strong>[</strong><strong>/</strong><strong>tex</strong><strong>]</strong></p><p></p><p></p><p><strong>[</strong><strong>tex</strong><strong>]</strong><strong> { \underline{ \boxed{\bf \therefore \: n = 7}}}$

Hence,

• No. of sides of polygon is 7.