Find Regular Polygon Sides from Interior Angle

This explanation will guide you through finding the number of sides of a regular polygon when you know the measure of each interior angle.

Understanding Interior and Exterior Angles

A regular polygon is a polygon where all sides are equal in length and all interior angles are equal in measure.

At each vertex of a polygon, the interior angle and the exterior angle form a linear pair, meaning their sum is always $180^\circ$.

We are given:

• Measure of each interior angle = $162^\circ$

Calculating the Exterior Angle

We can find the measure of each exterior angle using the relationship between interior and exterior angles:

Exterior Angle = $180^\circ$ - Interior Angle

Substitute the given interior angle value:

Exterior Angle = $180^\circ - 162^\circ$

Exterior Angle = $18^\circ$

Determining the Number of Sides

The sum of all exterior angles of any convex polygon is always $360^\circ$. For a regular polygon with $n$ sides, all exterior angles are equal. Therefore, the measure of each exterior angle can be calculated using the formula:

Exterior Angle = $\frac{360^\circ}{n}$

We know the exterior angle is $18^\circ$, so we can set up the equation to solve for $n$, the number of sides:

$18^\circ = \frac{360^\circ}{n}$

Solving for the Number of Sides (n)

To find $n$, we rearrange the equation:

$n = \frac{360^\circ}{18^\circ}$

Performing the division:

$n = 20$

Conclusion

Therefore, the regular polygon has 20 sides.