Question

Determine the number of sides of a polygon whose exterior and interior angles are in the ratio of $1:5$.

Answer 1

A polygon is a closed figure where the sides are all line segments.
A Polygon is a closed figure made up of lines segments (not curves) in two-dimensions.
A minimum of three-line segments are required for making a closed figure, thus a polygon with a minimum of three sides is known as Triangle.
Interior angle: An interior angle of a polygon is an angle inside the polygon at one of its vertices.

Exterior angle: An exterior angle of a polygon is an angle outside the polygon formed by one of its sides and the extension of an adjacent side.
Exterior angle of a polygon $= \dfrac{{{{360}^0}}}{{No.\,of\,sides\,of\,polygon}}$, and interior angle of polygon is $\dfrac{{\left( {n - 2} \right) \times {{180}^0}}}{n}$ (Where n is no. of sides).

Complete step by step solution:
Let the ratio of exterior and interior angle of polygon be $x$ .
Let exterior angle be $x$ and interior be $5x$.
We know that, each interior angle of a regular polygon $= {180^0} - (exterior\,angle)$

So,
$5x = {180^0} - x$
$5x + x = {180^0}$
$6x = {180^0}$
$x = \dfrac{{{{180}^0}}}{6}$
$x = {30^0}$.
So, exterior angle$= {30^0}$
& interior angle$= 5 \times {30^0}$
$= {150^0}$.
Now,
Exterior angle $= \dfrac{{{{360}^0}}}{{number\,of\,sides}}$
${30^0} = \dfrac{{{{360}^0}}}{{number\,sides}}$

Number of sides$\, = \dfrac{{{{360}^0}}}{{30}}$
$no.\,of\,sides = 12.$

Note: Given ratio of exterior and interior angle of polygon $1:5$.
Then, exterior angle is $\dfrac{1}{6} \times {180^0}$and interior angle is $\dfrac{5}{6} \times {180^0}$.
Exterior angle $= {30^0}$
Interior angle $= {150^0}$
Number of sides of polygon $= \dfrac{{{{360}^0}}}{{exterior\,angle}}$
$= \dfrac{{{{360}^0}}}{{30}} = 12$.