Question

If the smallest angle of rotation for a regular polygon is 18°, how many sides does polygon have? 10 12 20 24

Answer 1

Since the sum of the angles is 360°, we need to divide 360 with the number of sides that a polygon has.
360/18=20; this polygon has 20 sides.
Result:20

Answer 2

Answer:

The polygon has 20 sides (Icosagon)

Step-by-step explanation:

As the smallest angle of rotation for a regular polygon is 18°, so a 18° rotation yields the same figure.

According to the attachment, the side AB subtends an angle of 18° at the centre. That is why a 18° rotation generates the same figure.

According to the properties of regular polygon,

$OA=OB\ \ \Rightarrow m\angle OAB=m\angle OBA$

And

$m\angle OAC=m\angle OAB$

(as the radius of a regular polygon bisects the interior angle)

As in the triangle OAB,

$\Rightarrow m\angle AOB+m\angle OAB+m\angle OBA=180^{\circ}$

$\Rightarrow 18^{\circ}+2m\angle OAB=180^{\circ}$

$\Rightarrow m\angle OAB=81^{\circ}$

So,

$\Rightarrow m\angle CAB=2m\angle OAB=162^{\circ}$

So, interior angle of polygon is 162°

We know that, the interior angle of regular polygon with n side is,

$=\dfrac{(n-2)180}{n}$

Hence,

$\Rightarrow \dfrac{(n-2)180}{n}=162$

$\Rightarrow (n-2)180=162n$

$\Rightarrow 180n-360=162n$

$\Rightarrow 180n-162n=360$

$\Rightarrow 18n=360$

$\Rightarrow n=20$

Therefore, the polygon has 20 sides (Icosagon)