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Each exterior angle of a certain regular polygon measures $30{}^\circ .$ How many sides does the polygon have?

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Last updated date: 29th Feb 2024
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IVSAT 2024
Answer
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Hint: Here it is given that the polygon is a regular polygon.
As we know that the sum of all exterior angles of polygon is $360{}^\circ .$
But for regular polygons all sides of the polygon are equal.
That’s why there is only one measure of angle given in the question.
So, we have to only divide the sum of all exterior angles by the given angle for getting the number of sides of a given polygon.

Complete step by step solution:
We know that, the sum of all exterior angle of any regular polygon is always $360{}^\circ .$ Whether that quadrilateral polygon has $3$ sides or $16$ sides, its sum will always be $360{}^\circ .$
Given that, there is a regular polygon has measure of a angle is $30{}^\circ .$
So, for a regular polygon all exterior angles are equal and are of $30{}^\circ .$
As, the sum of all angles of regular polygon is $360{}^\circ .$ and one given angle is $30{}^\circ .$
The polygon has the total number of sides is $\dfrac{360{}^\circ }{30{}^\circ }=12.$

Hence, the given polygon has total $12$ sides.

Additional Information:
A closed-figure which is made up of the line segments and without any cures in the two dimensional plane is called a polygon. Polygon is made up of two words that are ‘poly’ and ‘gon’ which means many and sides respectively. For making a closed figure. We need at least three lines segments which connect from end to end. Therefore a polygon with at least three sides is called a Triangle.
There are some polygons which are made up of many numbers of lines. Which is as follows:
Triangle, Quadrilateral, Pentagon, Octagon, Hexagon, There are total $4$ types of polygon:
(1) Regular polygon has all interior angles and sides equal. E.g. square rhombus. Etc.
(2) Irregular polygon has all interior angles and sides different. E.g. rectangle, scalene triangle, etc.
(3) Convex polygon- has less than $180{}^\circ $ all interior angles.
(4) Concave polygon- has more than $180{}^\circ $ one or more than one interior angle.

Note: The angles of polygon are divided into two parts i.e.
(1) Interior angle
(2) Exterior angle
The sum of an interior angle of a polygon with four sides has $360{}^\circ .$
A polygon having exterior angles is the $360{}^\circ $ sum of all the exterior angles.
For regular polygons the measures of exterior angles are equal.
Hence the sum of all exterior angles for a polygon is $360{}^\circ .$