Question

Find the measure of each exterior angle of a regular polygon of
i) 9 sides
ii) 15 sides

Answer 1

Hint: First of all, we know that the sum of all the exterior angles of any polygon is ${360^ \circ }$. Take the measure of any angle in the polygon with 9 sides as $x$, then the sum of all the angles will be $9x$. Equate it to ${360^ \circ }$ and solve for the value of $x$. Similarly, the sum of all the angles of 15 sided polygon will be $15y$, where $y$ is the measure of each angle. Next, equate it to ${360^ \circ }$ to find the value of $y$.

Complete step by step solution: We are given regular polygons with 9 sides and 15 sides.
We have to find the measure of the exterior angle of the regular polygons.
We know that an exterior angle is an angle between any side and the line extended from the next side.
Also, the sum of exterior angles of any polygon is always ${360^ \circ }$.
Now, we will first find the measure of the exterior angle of a regular polygon with 9 sides.
If a polygon has 9 sides, it means it has 9 angles.
In a regular polygon, we know that all sides and angles are equal.
Hence, the sum of 9 exterior angles is ${360^ \circ }$

Let the measure of each angle be $x$, then we can say that,
$9x = {360^ \circ }$
Divide the equation throughout by 9
$x = {40^ \circ }$
Therefore, each exterior angle of the polygon with 9 sides has a measure of ${40^ \circ }$
Similarly, let each angle of the polygon with five sides as $y$
Then the sum of all the angles is ${360^ \circ }$, which implies
$15x = {360^ \circ }$
Divide the equation throughout by 15
$x = {24^ \circ }$
Therefore, each exterior angle of the polygon with 15 sides has a measure of ${24^ \circ }$

Note: A polygon has the number of angles equal to the number of sides in it. Also, the angles and sides are equal in the case of the regular polygon. The sum of the exterior angles of all the polygons is ${360^ \circ }$ whereas the sum of angles of interior polygons may vary.