Question

Each interior angle of a regular polygon is $ 18{}^\circ $ more than eight times an exterior angle. The number of sides of the polygon is:
A. 10
B. 15
C. 20
D. 25

Answer 1

Hint:Remember that a polygon has as many angles as its number of sides. The sum of the interior and exterior angle at a point / vertex is $ 180{}^\circ $ . Find the value of one exterior angle in degrees using the given relation. The sum of all the exterior angles of a polygon is always $ 360{}^\circ $ .

Complete step by step solution:
Let's say that there are n number of sides of the polygon and each exterior angle is of measure $
x{}^\circ $ .

Since, the sum of an interior angle and an exterior angle at a point is $ 180{}^\circ $ , we can say that:

Each exterior angle is $ 180{}^\circ $ − $ x{}^\circ $ .

According to the question, $ 180{}^\circ $ − $ x{}^\circ $ = 8 $ x{}^\circ $ + $ 18{}^\circ $ .
⇒ 9 $ x{}^\circ $ = $ 162{}^\circ $

Dividing both sides by 9, we get:
⇒ $ x{}^\circ $ = $ \dfrac{162{}^\circ }{9} $
⇒ $ x{}^\circ $ = $ 18{}^\circ $

Now, we also know that the sum of all the exterior angles of a polygon is always $ 360{}^\circ $ .
Therefore, n × $ 18{}^\circ $ = $ 360{}^\circ $ .

Dividing both sides by 9, we get:
⇒ n = $ \dfrac{360{}^\circ }{18} $
⇒ n = 20

The correct answer is C.

Note:The sum of all the interior angles of a n-sided polygon is always n × $ 180{}^\circ $ − $ 360{}^\circ $ .
A regular polygon is a polygon whose all sides as well as all angles are equal to each other.