Question

Find the number of sides of a regular polygon whose each exterior angle has a measure of
(i) \[60^\circ \]
(ii) \[72^\circ \].

Answer 1

Hint: We will use the fact that the sum of the exterior angles of a polygon is \[360^\circ \]. We will substitute the given measures of angles in the formula and solve it further to get the required number of sides. A polygon is of different shapes made of straight lines and can be drawn on paper.

Formula used:
We will use the formula Number of sides \[ \times \] Measure of each exterior angle \[ = 360^\circ \].

Complete step-by-step answer:
Let the number of sides of the polygon be \[n\]. It is given that the measure of each exterior angle of the polygon is \[60^\circ \].
Using the formula, Number of sides \[ \times \] Measure of each exterior angle\[ = 360^\circ \], we get
 \[n \times 60^\circ = 360^\circ \]
Dividing both sides by \[60^\circ \], we get
\[ \Rightarrow \dfrac{{n \times 60^\circ }}{{60^\circ }} = \dfrac{{360^\circ }}{{60^\circ }}\]
\[ \Rightarrow n = 6\]
$\therefore $ The number of sides of a polygon with each exterior angle \[60^\circ \] is 6.
Let the number of sides of the polygon be \[n\]. It is given that the measure of each exterior angle of the polygon is \[72^\circ \].
Using the formula, Number of sides \[ \times \] Measure of each exterior angle\[ = 360^\circ \], we get
\[n \times 72^\circ = 360^\circ \]
Dividing both sides by \[72^\circ \], we get
\[ \Rightarrow \dfrac{{n \times 72^\circ }}{{72^\circ }} = \dfrac{{360^\circ }}{{72^\circ }}\]
Thus, we get
\[n = 5\]
$\therefore $ The number of sides of a polygon with each exterior angle \[72^\circ \] is 5.

Note: The sum of exterior angles of a polygon is obtained by adding all the exterior angles attached to each side of the polygon. So, if a regular polygon has \[n\] sides with each exterior angle \[x^\circ \] then we add \[x^\circ \] ‘\[n\]’ times which is the same as multiplying \[x^\circ \] by \[n\]. This is the reason we have,
Sum of exterior angles \[ = \] Number of sides \[ \times \] Measure of each exterior angle
In all cases, the sum of exterior angles of a regular polygon is taken to be \[360^\circ \].