Question

The reation between an exterior angle and
interior angle of a regular polygon is 2.3.
Find the number of the sides in
and write the suitable name of palygon.
the polygon.
please answer me maths toppers, please ​

Answer 1

Answer:

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

So 2x + 3x = 180

5x = 180

x = 36

So exterior angle = 2x = 2 × 36 = 72°

So number of sides, n = 360/exterior angle

n = 360/72 = 5

Hence, its a pentagon.

Answer 2

Given :

• Ratio between the exterior angle and interior angle of a regular polygon = 2 : 3

To Find :

• The number of sides and name of the polygon

Solution :

Now , each interior angle of a regular polygon is given by ;

$\\ \star \: {\boxed{\purple{\sf{ Each \: interior \: angle_{(regular \: polygon)} = \dfrac{ {180}^{ \circ} (n - 2)}{n} }}}}$

Where ,

• n is number of sides of the polygon

Now , Each Exterior angle of a regular polygon is given by ;

$\\ \star \: {\boxed{\sf{\purple{Each \: exterior \: angle_{(regular \: polygon)} \: = \: \dfrac{ {360}^{ \circ} }{n} }}}}$

Where ,

• n is number of sides of the polygon

According to the question ,

$\\ : \implies \sf \: \dfrac{ {360}^{ \circ} }{n} : \dfrac{ {180}^{ \circ}(n - 2) }{n} = 2 : 3$

$\\ : \implies \sf \: \dfrac{360}{n} \times \frac{n}{180(n - 2)} = \dfrac{2}{3}$

$\\ : \implies \sf \: \dfrac{ 360 }{180(n - 2)} = \dfrac{2}{3}$

$\\ : \implies \sf \: {180}^{ \circ}(n - 2) \times 2 = 3 \times {360}^{ \circ}$

$\\ : \implies \sf \: 180 n - 360 \times 2= 1080$

$\\ : \implies \sf \: 180n - 360 = \dfrac{1080}{2}$

$\\ : \implies \sf \: 180n - 360 = 540$

$\\ : \implies \sf \: 180n = 540 + 360$

$\\ : \implies \sf \: 180n = 900$

$\\ : \implies{\underline{\boxed {\pink{\mathfrak{n = 5}}}}} \: \bigstar$

Hence ,

• The number of sides of the given polygon are 5 . So , the name of the given regular polygon is pentagon.