Question

If an interior angle of regular polygon measures\[{120^ \circ }\], how many sides does the polygon have?

Answer 1

Hint: Summation of interior and exterior angles of a regular polygon is \[{180^ \circ }\] and if the interior or exterior angle is given then by subtraction anyone from \[{180^ \circ }\]we can get the other one. And once a relationship between interior and exterior angle is formed then one can easily get the other angle which was asked.

Complete step by step answer:
Let the exterior angle of the regular polygon be ”y”
Now as we know the sum of exterior and interior angle is \[{180^ \circ }\]
So, the summation of the interior and exterior angle should be \[{180^ \circ }\]
\[
  y + {120^ \circ } = {180^ \circ } \\
  y = {180^ \circ } - {120^ \circ } \\
  y = {60^ \circ } \\
 \]
Now again another property used is summation of all the exterior angle of a polygon is \[{360^ \circ }\]
So,
\[{360^ \circ } \div {60^ \circ } = 6\]
Hence polygon has 6 sides.
Formulae Used: Polygon property is used i.e. summation of all exterior angles of a polygon is \[{360^ \circ }\]and the sum of interior and exterior angle of a polygon is\[{180^ \circ }\].
Additional Information: It is very easy to calculate such a question because the very basic property is used but you have to be careful because sometimes the question is a little bit twisted by giving the relations between the interior and exterior angle.

Note:
You can learn basic properties of polygon like the summation of all the exterior angles is \[{360^ \circ }\]and summation of interior and exterior angle of a polygon is\[{180^ \circ }\].
For a regular polygon, the length of all the sides is the same and has equal angles between the adjacent sides of the polygon. For irregular polygons, sides are not equal and even not have the same angles between the sides.