Let's find the number of sides of a regular polygon with an interior angle of 162°. Here are the steps:
1. Understand the Relationship:
The formula to calculate the interior angle [tex]\(\theta\)[/tex] of a regular polygon with [tex]\(n\)[/tex] sides is:
[tex]\[
\theta = \frac{(n-2) \times 180^\circ}{n}
\][/tex]
We are given that [tex]\(\theta = 162^\circ\)[/tex].
2. Set Up the Equation:
Substitute the given interior angle into the formula:
[tex]\[
162 = \frac{(n-2) \times 180}{n}
\][/tex]
3. Clear the Fraction:
Multiply both sides of the equation by [tex]\(n\)[/tex] to get rid of the denominator:
[tex]\[
162n = (n-2) \times 180
\][/tex]
4. Expand and Simplify:
Expand the right-hand side:
[tex]\[
162n = 180n - 360
\][/tex]
5. Isolate [tex]\(n\)[/tex]:
Move all terms involving [tex]\(n\)[/tex] to one side of the equation:
[tex]\[
162n - 180n = -360
\][/tex]
Simplify the left side:
[tex]\[
-18n = -360
\][/tex]
6. Solve for [tex]\(n\)[/tex]:
Divide both sides by -18:
[tex]\[
n = \frac{-360}{-18}
\][/tex]
[tex]\[
n = 20
\][/tex]
So, the number of sides of the regular polygon with an interior angle of 162° is [tex]\( \boxed{20} \)[/tex].
