To find the number of sides [tex]\( n \)[/tex] of a regular polygon given that one exterior angle measures [tex]\( 36^\circ \)[/tex], we can use the fact that the sum of all exterior angles in any polygon is always [tex]\( 360^\circ \)[/tex].
Here's a step-by-step explanation:
1. Understand the Property of Exterior Angles: The sum of all exterior angles of any polygon is [tex]\( 360^\circ \)[/tex]. This is a fundamental property that holds true regardless of the number of sides of the polygon.
2. Determine the Number of Exterior Angles: Since a regular polygon has all its exterior angles equal, and these exterior angles add up to [tex]\( 360^\circ \)[/tex], you can find the number of such angles by dividing [tex]\( 360^\circ \)[/tex] by the measure of one exterior angle.
3. Apply the Formula: If the measure of one exterior angle is given as [tex]\( 36^\circ \)[/tex], the number of sides [tex]\( n \)[/tex] can be found by:
[tex]\[ n = \frac{360^\circ}{36^\circ} \][/tex]
4. Perform the Division:
[tex]\[ n = \frac{360}{36} \][/tex]
[tex]\[ n = 10 \][/tex]
Therefore, the number of sides [tex]\( n \)[/tex] of the regular polygon is [tex]\( 10 \)[/tex].
