Question

Each exterior angle of a regular polygon measures\[\text{2}0{}^\circ \]. How many sides does the polygon have?

Answer 1

Hint: If we already have the each exterior angle of a regular polygon to find the number of sides of a regular polygon, we will follow the some steps written below as:
Firstly we will assume the number of sides.
Then we will use the formula of the sum of the exterior angles of a regular polygon that is:
\[\Rightarrow \text{Sum of the exterior angles of a regular Polygon }=\text{ number of sides}\times \text{each exterior angle}\]

Complete step-by-step answer:
Since, we have each exterior angle of a regular polygon \[\text{2}0{}^\circ \] that is given in the question.
So, here we will assume number of sides for given regular polygon as a variable:
Let’s consider that the number of sides of a regular polygon is $n$ .
As we know that the sum of the all exterior angles of a regular polygon is equal to $360{}^\circ $ and we know that the sum of the all exterior angles of a regular polygon is also equal to the multiplication of number of sides and each exterior angle of a regular polygon. So, we will use this formula as:
\[\Rightarrow \text{Sum of the exterior angles of a regular Polygon }=\text{ number of sides}\times \text{each exterior angle}\]
Now, we will apply the given value of each exterior angle and other values in the above formula as:
\[\Rightarrow \text{360}{}^\circ \text{ }=\text{ n}\times \text{20}{}^\circ \]
Now, we will combine the numbers one side as;
\[\Rightarrow \text{n}=\text{ }\dfrac{\text{360}{}^\circ }{\text{20}{}^\circ }\]
That describes that we will have to use division in the above equation as:
\[\Rightarrow \text{n}=\text{ 360}{}^\circ \div 20{}^\circ \]
Here, we will get $18$ as a result of the above equation as:
\[\Rightarrow \text{n}=\text{ 18}\]
Hence, the number of the sides of the given regular polygon is \[\text{18}\] .

Note:Since, we know that the sum of the all exterior angles of a regular polygon is equal to $360{}^\circ $. So, we can check that our solution is correct or not by putting the numbers of sides in the formula that is
\[\Rightarrow \text{Sum of the exterior angles of a regular Polygon }=\text{ number of sides}\times \text{each exterior angle}\]
Here, we have a number of sides and each exterior angle of a regular polygon. So we will apply them in the above formula as:
\[\Rightarrow \text{Sum of the exterior angles of a regular Polygon }=\text{ 18}\times \text{20}{}^\circ \]
\[\Rightarrow \text{Sum of the exterior angles of a regular Polygon }=\text{ 360}{}^\circ \]
Here, we got that the sum of the all exterior angles of a regular polygon is equal to $360{}^\circ $by applying the values. Hence, the solution is correct.