Question

If each interior angle of a regular polygon is $10$ times its exterior angle, then the number of sides of polygon is
$
  A.{\text{ 12}} \\
  {\text{B}}{\text{. 18}} \\
  {\text{C}}{\text{. 22}} \\
  {\text{D}}{\text{. 24}} \\
 $

Answer 1

Hint:The polygon is the plane shaped two dimensional shape with finite number of the straight line segments to form the closed polygonal chain or the polygon circuit. For example triangles, rectangles and pentagons are polygons. The circle is not a polygon because it has the curved side.

Complete answer:
Let n be the number of the sides of the regular polygon.
Let us denote exterior be Ae
Therefore, Exterior angle Ae is $ = \dfrac{{360}}{n}$
And let the interior angle be Ai
Therefore, Exterior angle Ai is $ = \dfrac{{n - 2}}{n} \times 180$
Given that - Each interior angle of a regular polygon is $10$ times its exterior angle
$
  {A_i} = 10 \times {A_e}
  \therefore \dfrac{{n - 2}}{n} \times 180 = 10 \times \dfrac{{360}}{n}
 $
As, same number from both the sides of the equations cancels each other, here “n” is cancelled from both the sides.
$180(n - 2) = 10 \times 360$
Also, the numerator in the multiplicative part of the left hand side goes to the denominator of the right hand side.
$n - 2 = \dfrac{{3600}}{{180}}$
Simplify the above equation –
$n - 2 = 20$
When the number changes its sides, sign of the number changes from positive to negative and vice-t
$
  \therefore n = 20 + 2
  \therefore n = 22
 $
Therefore, if each interior angle of a regular polygon is $10$ times its exterior angle, then the number of sides of polygon is $22$
Hence, from the given multiple choices, option C is the correct answer.

Note:The interior part of the solid polygon is also called its body. It is entitled on the basis of the number of sides of the polygon. For example – An n-gon is a polygon with n sides; a triangle is the polygon with $3 - gon$.