Question

The exterior angle of a regular polygon is one-third of its interior angle . How many sides has the polygon got

Answer 1

GIVEN :-

• The exterior angle of a regular polygon is one-third of its interior angle

TO FIND :-

• Sides of the regular polygon .

SOLUTION :-

  Let ,

  Interior angle = x

  Exterior angle = $\sf\dfrac{1}{3} x$

  No of side = n

 Sum of an exterior angle and an interior angle = 180

 $\longrightarrow\sf x+\dfrac{1}{3}x = 180 \\\\\longrightarrow \dfrac{3x+x}{3}= 180 \\\\\longrightarrow 4x= 180\times 3 \\\\\longrightarrow 4x= 540 \\\\\longrightarrow \bf x = 135$

 Interior angle = 135°

 Exterior angle = $\dfrac{1}{3}\sf x$

                        = $\sf \dfrac{1}{3} \times 135$

                        = 45°

 Each exterior angle = $\sf\dfrac{360}{n}$

  $\longrightarrow\sf 45 =\dfrac{360}{n}\\\\\longrightarrow 45n = 360 \\\\\longrightarrow\bf n = 8$

 Number of sides of the regular polygon = 8