Question

The ratio of an external angle and an internal angle of a regular polygon is 1:5. The number of sides in the polygon is
\[\begin{gathered}
  A.\;\;\;\;\;12 \\
  B.\;\;\;\;\;6 \\
  C.\;\;\;\;\;8 \\
  D.\;\;\;\;\;10 \\
\end{gathered} \]

Answer 1

Hint: In this question firstly we assume the number of sides of a given polygon be n. As we have given the ratio of internal and external angle we use the formula of internal and external angle for n sided polygon and equate them to get the desired result.

Complete step-by-step answer:
We have given the ratio of an external angle and an internal angle of a regular polygon is 1:5
To find the number of sides of a polygon let us suppose the polygon has ‘n’ number of sides.
As we all know the formula for an exterior angle of a polygon is \[\dfrac{{360}}{n};\] where n is the number of sides of a polygon.
Also, the formula for an interior angle of a polygon is \[\dfrac{{\left( {n - 2} \right)180}}{n};\] where n is the number of sides of a polygon.
Using the above facts we can write,
\[\dfrac{1}{5} = \dfrac{{\dfrac{{360}}{n}}}{{\dfrac{{\left( {n - 2} \right)180}}{n}}}\]
\[ \Rightarrow \dfrac{1}{5} = \dfrac{{360 \times n}}{{\left( {n - 2} \right) \times n \times 180}}\]
On further simplifying the above expression we get,
\[\dfrac{1}{5} = \dfrac{{360}}{{\left( {n - 2} \right) \times 180}}\]
\[ \Rightarrow \dfrac{1}{5} = \dfrac{2}{{\left( {n - 2} \right)}}\]
\[ \Rightarrow \left( {n - 2} \right) = 2 \times 5\]
\[ \Rightarrow n = 10 + 2\]
\[ \Rightarrow n = 12\]
Thus, the total number of sides in the polygon is 12.
Hence, option A. 12 is the correct answer.

Note: Interior angle of a polygon is an angle inside a polygon at a vertex of the polygon.
The sum of the interior angles of a simple n-polygon is \[\left( {n - 2} \right)180\]degrees.
Exterior angle of a polygon is an angle at a vertex of the polygon, outside the polygon, formed by one side and the extension of an adjacent side.
The sum of exterior angles of a polygon must be\[{360^ \circ }\].
The formula for calculating the size of an exterior angle is \[\dfrac{{360}}{n}\], where n is the number of sides of a polygon.