Question

An exterior angle and the interior angle of a regular polygon are in the ratio 2:7 .find the number of sides of the polygon​

Answer 1

Answer :-

The number of sides of polygon is 9.

Step-by-step explanation:

To Find :-

• The number of sides of the polygon.

★ Solution

Given that,

• The exterior angle and Interior angle of a regular polygon = 2:7

Let us assume the ratio exterior angle and Interior angle of the polygon as 2x and 7x respectively.

∴ 2x + 7x = 180°

⇒ 2x + 7x = 180

⇒ 9x = 180

⇒ x = 180/9

⇒ x = 20

The value of x is 20.

The exterior and interior angles are :-

• 2x = 2*20 = 40° ... exterior angle
• 7x = 7*20 = 140° ... interior angle

Now, A . T . Q,

• Number of sides are :-

⇒ 360/Measure of interior angle

⇒ 360/40

⇒ 36/4

⇒ 9 sides

Hence, Number of side of the regular polygon is 9 sides.

Answer 2

Answer:

Given :-

• An exterior angle and the interior angle of a regular polygon are in the ratio of 2 : 7.

To Find :-

• How many numbers of sides are in the polygon.

Solution :-

Let,

$\mapsto$ Exterior angle = 2x

$\mapsto$ Interior angle = 7x

As we know that :

$\bigstar$ Sum of exterior angle + interior angles of Polygon = 180°

According to the question by using the formula we get,

$\implies\sf 2x + 7x =\: 180^{\circ}$

$\implies \sf 9x =\: 180^{\circ}$

$\implies \sf x =\: \dfrac{\cancel{180^{\circ}}}{\cancel{9}}$

$\implies \sf x =\: \dfrac{20^{\circ}}{1}$

$\implies \sf \bold{\green{x =\: 20^{\circ}}}$

Hence, the required exterior angle and interior angle will be :

$\Rightarrow$ Exterior Angle :

$\mapsto \sf 2 \times 20^{\circ}$

$\mapsto\sf\bold{\purple{40^{\circ}}}$

$\Rightarrow$ Interior Angle :

$\mapsto \sf 7 \times 20^{\circ}$

$\mapsto \sf\bold{\purple{140^{\circ}}}$

Now, we have to find how many numbers of sides of the polygon :

As we know that :

$\footnotesize\longmapsto \sf\boxed{\bold{\pink{Each\: Interior\: Angle\: Of\: Regular\: Polygon =\: \dfrac{(n - 2) \times 180^{\circ}}{n}}}}$

where,

• n = Number of sides of the polygon

Given :

• Each interior angle of regular polygon = 140°

According to the question by using the formula we get,

$\longrightarrow \sf 140^{\circ} =\: \dfrac{(n - 2) \times 180^{\circ}}{n}$

$\longrightarrow \sf 140^{\circ} =\: \dfrac{180^{\circ}n - 360^{\circ}}{n}$

By doing cross multiplication we get,

$\longrightarrow \sf 180^{\circ}n - 360^{\circ} =\: 140^{\circ}n$

$\longrightarrow \sf 180^{\circ}n - 140^{\circ}n =\: 360^{\circ}$

$\longrightarrow \sf 40^{\circ}n =\: 360^{\circ}$

$\longrightarrow \sf n =\: \dfrac{\cancel{360^{\circ}}}{\cancel{40^{\circ}}}$

$\longrightarrow \sf\bold{\red{n =\: 9}}$

$\therefore$ The number of sides of the polygon is 9 sides .