To determine the number of sides [tex]\(n\)[/tex] of a regular polygon when given that one exterior angle measures [tex]\(20^\circ\)[/tex], we can use the relationship between the exterior angle and the number of sides of a regular polygon.
### Step-by-Step Solution:
1. Understand the Relationship:
- The sum of all exterior angles of any polygon is always [tex]\(360^\circ\)[/tex].
- In a regular polygon, all exterior angles are equal.
2. Formulate the Equation:
- Let [tex]\( \theta \)[/tex] be the measure of each exterior angle. For this problem, [tex]\( \theta = 20^\circ \)[/tex].
- The number of sides [tex]\(n\)[/tex] of the polygon can be found using the formula for the measure of an exterior angle in a regular polygon: [tex]\( \theta = \frac{360^\circ}{n} \)[/tex].
3. Solve for [tex]\(n\)[/tex]:
- Substitute the given exterior angle into the equation: [tex]\( 20^\circ = \frac{360^\circ}{n} \)[/tex].
- Rearrange the equation to solve for [tex]\(n\)[/tex]:
[tex]\[
n = \frac{360^\circ}{20^\circ}
\][/tex]
4. Calculate the Values:
- Perform the division:
[tex]\[
n = \frac{360}{20} = 18
\][/tex]
Therefore, the regular polygon has [tex]\( \boxed{18} \)[/tex] sides.
