13 Find the number of sides of a regular polygon with interior angle $$ 162^{\circ} $$.

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To determine the number of sides $$ n $$ of a regular polygon with each interior angle measuring $$ 162^\circ $$, you can use the formula for the interior angle of a regular polygon:

$$
\text{Interior Angle} = \frac{(n - 2) \times 180^\circ}{n}
$$

Given that the interior angle is $$ 162^\circ $$, set up the equation:

$$
\frac{(n - 2) \times 180^\circ}{n} = 162^\circ
$$

Now, solve for $$ n $$:

1. Multiply both sides by $$ n $$:

$$
(n - 2) \times 180^\circ = 162^\circ \times n
$$

2. Distribute $$ 180^\circ $$ on the left side:

$$
180n - 360 = 162n
$$

3. Subtract $$ 162n $$ from both sides:

$$
18n - 360 = 0
$$

4. Add $$ 360 $$ to both sides:

$$
18n = 360
$$

5. Finally, divide both sides by $$ 18 $$:

$$
n = \frac{360}{18} = 20
$$

**Answer:** 20
A regular polygon with an interior angle of $$ 162^\circ $$ has 20 sides.

13 Find the number of sides of a regular polygon with interior angle \( 162^{\circ} \).

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