Question

How many sides are in a regular polygon that has an exterior angle of 40 degrees?

Answer 1

Hint: We will first mention the formula which relates the exterior angle and the number of sides of a regular polygon and then put in the given exterior angle and get the required answer.

Complete step-by-step answer:
We are given that we need to find the number of sides in a regular polygon if the exterior angle is given to be 40 degrees.
We know that the formula of number of sides of a regular polygon is given by the following expression:-
$ \Rightarrow n = \dfrac{{360}}{\theta }$, where n is the number of sides of a regular polygon and $\theta $ is the exterior angle in terms of degrees.
Now, we are given that the exterior angle if 40 degrees, let us put this in the above mentioned formula to get:-
$ \Rightarrow n = \dfrac{{360}}{{40}}$
Cancelling off 0 from both the numerator and denominator on the right hand side so that we obtain the following information:-
$ \Rightarrow n = \dfrac{{36}}{4}$
Now, we can simply do the required calculations and get the following required answer:-
$ \Rightarrow n = 9$

Hence, the regular polygon which has an exterior angle of 40 degrees has 9 sides in it.

Note:
The students must note that they must commit to memory the following formula:-
$ \Rightarrow n = \dfrac{{360}}{\theta }$, where n is the number of sides of a regular polygon and $\theta $ is the exterior angle in terms of degrees.
The students must also take care of the fact that they either need to change both the numerator and denominator of the formula in radians instead of degrees because whatever we are putting in both of them should be in the same units and not differing ones. Because if you put one angle in degrees and one in radians, that will be a mistake.
Let us write the same formula in terms of radians, it will be as follows:-
$ \Rightarrow n = \dfrac{{2\pi }}{\theta }$, where n is the number of sides of a regular polygon and $\theta $ is the exterior angle in terms of radians.