Question

The exterior angle of a regular polygon is one-third of its interior angle. Find the no of sides

Answer 1

If x is the measure of the exterior angle, then the measure of the interior angle is 3x. Then your equation is x + 3x = 180 degs, which implies x = 45 degs. Thus the exterior angle is 45 degs. Then the number of sides is n = 8

Answer 2

Answer:

${\huge{\underline{\underline{\sf{\blue{ Question:-}}}}}}$

The exterior angle of a regular polygon is one-third of its interior angle. Find the no. of sides.

${\huge{\underline{\underline{\sf{\blue{ Solution:-}}}}}}$

Hence ,

Let number of sides of regular polygon = n

Each exterior angle

$Each \: exterior \: angle \: = \: ( \frac{360}{n} )°</p><p>$

$Each \: exterior \: angle \: = \: ( \frac{2(n - 2)}{n} \times 90 \: )$

The exterior angle is one - third of its interior angle.

Therefore

$Therefore \: \frac{360}{n } \: = \: \frac{1}{3} ( \frac{2n - 4}{n} \times 90)$

$= > \: \: \: \: \: \frac{360}{n} = \frac{2n - 4}{n} \times 30$

$= > \: \: \: \: \: \: 360 = 60n - 120$

$= > \: \: \: 360 + 120 = 60n$

$= > \: \: \: 480 = 60n$

$= > \: \: \: \: n \: = \frac{480}{60} = 8$

Thus,

The number of sides of regular polygon = 8 .

${\huge{\underline{\underline{\sf{\blue{Alternate\: Method }}}}}}$

Let exterior angle = x

So, interior angle = 3x

Since , x + 3x = 180° (linear pair )

$= > 4x = 180 \: °</p><p>= > x = \frac{180 \: °</p><p>}{4} = 45 \: °</p><p>$

Let n be the number of sides of the regular polygon.

Then , n × 45° = 360 °

(Exterior angle sum property )

$So, \: n = \frac{360 \:°}{45 \: °</p><p>} = 8$

Thus ,

The number of the sides of regular polygon = 8.