Question

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

Answer 1

Hint: 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 \[ = \dfrac{{{{360}^0}}}{{exterior\,angle}}\]
\[ = \dfrac{{{{360}^0}}}{{30}} = 12\].