Question

The ratio between exterior and interior angle of a regular polygon is 1:4. Find the number of sides of the polygon.

Answer 1

Hint: We start solving the problem by recalling the properties of regular polygon and assigning the variables for the interior and exterior angle of given regular polygon. We then use the given ratio between exterior and interior angles to get the relation between them. We use this relation and the property of the sum of interior and exterior angle in a polygon is ${{180}^{\circ }}$ to get the value of interior and exterior angles. We then use the property that the sum of all exterior angles in a polygon is ${{360}^{\circ }}$ to get the number of sides of a polygon.

Complete step-by-step answer:
According to the problem, we have given that the ratio between exterior and interior angle of a regular polygon is $1:4$.
Before solving for the number of sides, we recall the properties of regular polygons. We know that a regular polygon is a polygon that is equiangular, where all angles are equal in measure and equilateral, all sides have the same length.
Let us take the interior angle of the regular polygon as ‘x’ and exterior angle as ‘y’. According to the problem, we have the ratio of y and x as $1:4$.
So, we get \[\dfrac{y}{x}=\dfrac{1}{4}\].
\[\Rightarrow y=\dfrac{x}{4}\] ---(1).
We know that the sum of exterior angle and interior angle in a polygon is ${{180}^{\circ }}$.
So, we get $x+y={{180}^{\circ }}$.
From equation (1), we get $x+\dfrac{x}{4}={{180}^{\circ }}$.
$\Rightarrow \dfrac{4x+x}{4}={{180}^{\circ }}$.
$\Rightarrow \dfrac{5x}{4}={{180}^{\circ }}$.
$\Rightarrow 5x=4\times {{180}^{\circ }}$.
$\Rightarrow 5x={{720}^{\circ }}$.
$\Rightarrow x=\dfrac{{{720}^{\circ }}}{5}$.
$\Rightarrow x={{144}^{\circ }}$.
So, we have got the interior angle x as ${{144}^{\circ }}$.
Let us now find the exterior angle of the given polygon from equation (1).
So, we have an external angle $y=\dfrac{{{144}^{\circ }}}{4}$.
$\Rightarrow y={{36}^{\circ }}$ ---(2).
So, we have got the exterior angle as ${{36}^{\circ }}$.
Let us assume the number of sides of the given polynomial as $n$. We know that the sum of the $n$ external angles in the polygon is ${{360}^{\circ }}$.
So, we get $n\times \left( \text{exterior angle} \right)={{360}^{\circ }}$.
From equation (2), we get $n\times {{36}^{\circ }}={{360}^{\circ }}$.
$\Rightarrow n=\dfrac{{{360}^{\circ }}}{{{36}^{\circ }}}$.
$\Rightarrow n=10$.
So, we have found the number of sides of a given regular polygon as 10.
∴ The number of sides of a given regular polygon is 10.

Note: We should keep in mind that the sum of the exterior angles in any polygon is ${{360}^{\circ }}$. This can be verified by checking the exterior angles of triangle, square etc. We can also find the sum of the interior angles of the regular polygon by using sum of the interior angle as $\left( n-2 \right)\times {{180}^{\circ }}$ after finding the total no. of sides of the regular polygon. Similarly, we can expect problems to find all the exterior angles for irregular polygons.