Question

If an exterior angle of a regular polygon measures \[{{120}^{\circ }}\]. How many sides does the polygon have?

Answer 1

Hint: In this question we have to find the number of sides of a polygon. As we know that all the exterior angles of a polygon add up to \[{{360}^{\circ }}\]. So, each exterior angle is \[\dfrac{{{360}^{\circ }}}{n}\]. (Where \[n\]is the number of sides.) We will use this method in order to solve the question, as we have given the measure of each exterior angle is \[{{120}^{\circ }}\].

Complete Step by Step solution:
As we know that the sum of exterior angles of a polygon is \[{{360}^{\circ }}\].
and the formula for calculating the size of exterior angle of a polygon is given by each exterior angle \[=\dfrac{{{360}^{\circ }}}{n}\]; \[n\]is the numbers of sides of the polygon. ..............(i)
So, here in the question we have given an exterior angle of a regular polygon is \[{{120}^{\circ }}\].
Which implies that each exterior angle of a required polygon is of measure\[{{120}^{\circ }}\].
\[\therefore \] formula \[\left( i \right)\] will become
\[{{120}^{\circ }}=\dfrac{{{360}^{\circ }}}{number\,of\,sides\,of\,polygon\,\left( n \right)}\]
\[\Rightarrow \]number of sides of a regular polygon \[=\dfrac{{{360}^{\circ }}}{{{120}^{\circ }}}\]
\[\Rightarrow \]number of sides of a regular polygon \[=3\]
\[\Rightarrow \] If an exterior angle of a regular polygon is \[{{120}^{\circ }}\] then the regular polygon will have \[3\] number of sides.

\[\Rightarrow \]The regular polygon is a triangle.

Note:
First of all we will understand what a regular polygon is.
Definition of Regular polygon:
A regular polygon is simply a polygon whose all sides have the same length and angles have the same measure.
Examples of regular polygon:
Regular hexagon, regular pentagon, in equilateral triangle.
Definition of Exterior angles: An exterior angle is an angle formed by one side of a simple polygon and a line extended from on adjacent sides.
Definition of interior angle: An interior angle is an angle if a point within the angle is in the interior of the polygon.
Some key points to remember about internal and external angles.
-A polygon has exactly one internal angle per vertex.
-The sum of the internal angle and exterior angle on the same vertex is \[{{180}^{\circ }}\].
-If every internal angle of a simple polygon is less than \[{{180}^{\circ }}\] then the polygon is convex.
-Sum of interior angles \[=\left( n-2 \right)\times {{180}^{\circ }}\]; where \[n\] is the number of sides of a regular polygon.
-each interior angle (of a regular polygon) \[=\dfrac{\left( n-2 \right)\times {{180}^{\circ }}}{n}\]. Where \[n\]is the number of sides of a regular polygon.