Question
What is the sum of the exterior angles of every polygon?
If the exterior angle of a regular polygon measures 40°, how many sides does the polygon have?
If the exterior angle of a regular polygon measures 40°, how many sides does the polygon have?
Answers
Answered by
GPT-4o mini
The sum of the exterior angles of any polygon is always \(360^\circ\), regardless of the number of sides the polygon has.
Now, to find the number of sides of a regular polygon with an exterior angle of \(40^\circ\), we can use the fact that the measure of each exterior angle of a regular polygon is given by the formula:
\[
\text{Exterior angle} = \frac{360^\circ}{n}
\]
where \(n\) is the number of sides.
Setting the exterior angle equal to \(40^\circ\):
\[
40^\circ = \frac{360^\circ}{n}
\]
To find \(n\), we can rearrange the equation:
\[
n = \frac{360^\circ}{40^\circ}
\]
Calculating this gives:
\[
n = 9
\]
Thus, the polygon is a nonagon (9-sided polygon). Therefore, the number of sides of the polygon is \(\boxed{9}\).
Now, to find the number of sides of a regular polygon with an exterior angle of \(40^\circ\), we can use the fact that the measure of each exterior angle of a regular polygon is given by the formula:
\[
\text{Exterior angle} = \frac{360^\circ}{n}
\]
where \(n\) is the number of sides.
Setting the exterior angle equal to \(40^\circ\):
\[
40^\circ = \frac{360^\circ}{n}
\]
To find \(n\), we can rearrange the equation:
\[
n = \frac{360^\circ}{40^\circ}
\]
Calculating this gives:
\[
n = 9
\]
Thus, the polygon is a nonagon (9-sided polygon). Therefore, the number of sides of the polygon is \(\boxed{9}\).