To find the number of sides [tex]\( n \)[/tex] of a regular polygon given that each interior angle is [tex]\( 162^\circ \)[/tex], we can use the relationship between the interior angles and the number of sides of a polygon. Here's the step-by-step solution:
1. Formula for Interior Angle:
The formula to calculate the interior angle ([tex]\( A \)[/tex]) of a regular polygon with [tex]\( n \)[/tex] sides is:
[tex]\[
A = \frac{(n-2) \times 180^\circ}{n}
\][/tex]
2. Given Information:
We are given that the interior angle [tex]\( A \)[/tex] is [tex]\( 162^\circ \)[/tex].
3. Set Up the Equation:
Substitute [tex]\( A \)[/tex] with [tex]\( 162 \)[/tex] in the formula:
[tex]\[
162 = \frac{(n-2) \times 180^\circ}{n}
\][/tex]
4. Solve for [tex]\( n \)[/tex]:
To solve this equation, we first clear the fraction by multiplying both sides by [tex]\( n \)[/tex]:
[tex]\[
162n = (n-2) \times 180
\][/tex]
5. Distribute and Simplify:
Expand the right side of the equation:
[tex]\[
162n = 180n - 360
\][/tex]
6. Rearrange the Equation:
To isolate [tex]\( n \)[/tex], subtract [tex]\( 180n \)[/tex] from both sides:
[tex]\[
162n - 180n = -360
\][/tex]
Simplify the left side:
[tex]\[
-18n = -360
\][/tex]
7. Solve for [tex]\( n \)[/tex]:
Divide both sides by [tex]\(-18\)[/tex]:
[tex]\[
n = \frac{360}{18}
\][/tex]
8. Final Calculation:
[tex]\[
n = 20
\][/tex]
Therefore, the number of sides [tex]\( n \)[/tex] of the regular polygon is [tex]\( 20 \)[/tex].
