Question

A regular polygon has $21$ sides. Find the size of each interior angle?

Answer 1

Hint: Here, in the given question, we are given that a regular polygon has $21$ sides and we need to find the size of each interior angle. A polygon is a closed geometric figure which has only two dimensions (length and width). A regular polygon is a polygon whose sides are of equal length. The measure of each interior angle of a regular polygon is equal to the sum of interior angles of a regular polygon divided by the number of sides. So, at first we will find the sum of interior angles of a regular 21-gon using the $\left( {n - 2} \right) \times 180^\circ $ formula, where $n$ is the number of sides. After this, we will find the size of each interior angle.

Complete step-by-step answer:
The sum of interior angles of a regular 21-gon = $\left( {n - 2} \right) \times 180^\circ $ where, $n$ is the number of sides.
We are given that $n = 21$. Therefore, we get
$ = \left( {21 - 2} \right) \times 180^\circ $
On subtraction of terms, we get
$ = \left( {19} \right) \times 180^\circ $
On multiplication of terms, we get
$ = 3420^\circ $
Thus, the sum of interior angles of a regular polygon is $3420^\circ $. Now we will find the size of each angle.
As we know, the measure of each interior angle of a regular polygon is equal to the sum of interior angles of a regular polygon divided by the number of sides. Therefore, we get
$ = \dfrac{{3420}}{{21}}$
$ = 162.8^\circ $
Therefore, the size of each interior angle of 21-gon is $162.8^\circ $.

Note: Remember that to find the interior angle we used the $\left( {n - 2} \right) \times 180^\circ $ formula. But to find the exterior angle we can use this. We can find the exterior angle using the $\dfrac{{360^\circ }}{n}$ formula, where $n$ is the number of sides. We know that angle on a straight line is equal to $180^\circ $. So, if the interior angle is given we can find the exterior angle and vice-versa.