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The measure of the exterior angle regular polygon is 30 degrees. How many sides does it have?

Last updated date: 22nd Jul 2024
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Hint: We are given a measurement of the exterior angle of a regular polygon. We are asked to find number of side, we start with understanding what are polygon these understand the regular polygon then we find the relation between exterior angle and the number of side then we use exterior angle as ${{30}^{\circ }}$ and calculate the number of side.

Complete step by step answer:
We are given that a regular polygon has an exterior angle as ${{30}^{\circ }}$ , we have to find the number of sides of their polygon.
Before we start solving this we will learn that any closed figure with 3 or more sides is called polygon. Here poly means many. If it has 3 sides we call such a polygon as a triangle, if sides are 4 then we call them quadrilateral.
Now, if all the sides of any polygon are the same, all the angles are the same then such polygons are referred to as regular polygon so our polygon is regular. It means its all sides and all angles are equal.
Now to find the number of sides, we must know how the exterior angle and number of sides are connected.
In regular polygon the sum of all exterior angle is always ${{360}^{\circ }}$
So the measure of one angle is given as ${{360}^{\circ }}$ divided by the number of sides.
That is –
Exterior angle $=\dfrac{{{360}^{\circ }}}{n}$
Where, n = number of sides.
So as exterior angle is ${{30}^{\circ }}$
So we get –
$30=\dfrac{{{360}^{\circ }}}{n}$
By simplifying, we get –
$n=\dfrac{{{360}^{\circ }}}{30}=12$
So, number of sides are 12

There are more formulas related to polygon.
1. Sum of interior angle of polygon with n side is given as ${{180}^{\circ }}\left( n-2 \right)$
2. Number of diagonal in an sided polygon is given as –
$\dfrac{\left[ n\left( n-3 \right) \right]}{2}$
3. The measurement of interior angle of n sided polygon is $\dfrac{\left[ \left( n-2 \right){{180}^{\circ }} \right]}{n}$