Question

An exterior angle of a regular polygon is 15 degrees. How many sides does the polygon have?

Answer 1

Hint: We are given a regular polygon whose exterior angle is 15 degrees and we have to find the sides of the polygon. We know that the sum of the exterior angles of a polygon is $${360}^{\circ }$$. So, if we divide the total sum of the exterior angle of a polygon by the exterior angle, then, we will get the value of the number of sides of the polygon.

Complete step by step solution:
According to the question given to us, we are given the value of an exterior angle of a polygon. The value is 15 degrees. And using that, we have to find the number of sides of that polygon.
Polygon refers to the geometrical shape made using a finite number of straight lines and which are closed.
Example – Square, Triangle
We know that the total sum of exterior angles of any polygon is $${360}^{\circ }$$.
And if we want to know the exterior angle of any polygon, we divide $${360}^{\circ }$$ by the number of sides, ‘n’ of that particular polygon.
That is, $${\displaystyle \frac{{360}^{\circ }}{n}}$$.
For example – Square has 4 sides, so the value of exterior angle is $${\displaystyle \frac{{360}^{\circ }}{4}}={90}^{\circ }$$
So, if we want to find the number of sides then, we will divide $${360}^{\circ }$$ by the exterior angle.
We have,
$${\displaystyle \frac{{360}^{\circ }}{n}}={15}^{\circ }$$
Rearranging the above equation in terms of ‘n’, we have,
$$\Rightarrow {\displaystyle \frac{{360}^{\circ }}{{15}^{\circ }}}=n$$
$$\Rightarrow n={\displaystyle \frac{{360}^{\circ }}{{15}^{\circ }}}$$
Solving the above expression, we get the value as,
$$\Rightarrow n=24$$

Therefore, the number of sides of the polygon is 24 sides.

Note: The formula used for finding the number of sides should not be confused and should be used appropriately. The calculations should be done correctly and step-wise. And also, the exterior angle should not be confused with the interior angle, both are different in every aspect.
Sum of exterior angles of any polygon is $${360}^{\circ }$$, but the sum of interior angles is different for different polygon and can be calculated by using the formula $$(n-2)\times {180}^{\circ }$$, where $$n$$ is the number of sides of that polygon.