Question

The ratio of the interior angle and exterior angle of a regular polygon is 3:2. Find the number of sides in the polygon.​

Answer 1

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Given :

ratio of exterior angle and interior angle of a regular polygon = 2 : 3

Let exterior angle be 2x and interior angle be 3x.

We know that , each interior angle of a regular polygon = 180° - (exterior angle)

So, 3x = 180° - 2x

5x = 180°

x = 36°,

So, exterior angle = 2*36 = 72°

We know that each exterior angle of a regular polygon = (360/no. of sides of polygon)°

So , 72° = (360/ no.of sides)°

No. Of sides = 360/72 = 5.

Answer 2

Given,

The ratio of interior angle and exterior angle of a regular polygon is $3:2$.

Interior angle : Exterior angle $=3:2$.

According to the given ratio:

Let the interior angle be $3x$ and exterior angle be $2x$ of a regular polygon.

We know that, the sum of interior angle and its corresponding exterior angle is equal to $180 ^{o}$.

So, $(3x+2x)=180^{o}$

$5x=180^{o}$

$x=\frac {180^{o} } {5}$

$x=36^{o}$.

Then, interior angle of a regular polygon is $3x=(3\times36)=108^{o}$.

The exterior angle is $2x=(2\times36)=72^{o}$.

We know that, each exterior angle of a regular polygon$=\frac {360^{o} } {Number of sides}$

$72^{o} =\frac {360^{o} } {Number of sides}$

Number of sides$=\frac {360^{o} } {72^{o} }$

Number of sides$=5$.

Hence, number of sides in the given regular polygon is $5$.