一个正多边形有 $n$ 条边。当边数翻倍时，其每个内角增加 30°。求 $n$ 的值。 答案 n = ...

1. Write expressions for the interior angles before and after doubling the number of sides.

For a regular polygon with $n$ sides, each interior angle is
$\frac{(n-2)180^\circ}{n}.$
When the number of sides is doubled to $2n$, each interior angle is
$\frac{(2n-2)180^\circ}{2n}.$

2. Form an equation using the increase of $30^\circ$.

Since the new interior angle is $30^\circ$ larger,
$\frac{(2n-2)180}{2n}=\frac{(n-2)180}{n}+30.$

3. Solve the equation for $n$.

Simplify:
$\frac{(2n-2)180}{2n}=180-\frac{180}{n}$
and
$\frac{(n-2)180}{n}=180-\frac{360}{n}.$
So
$180-\frac{180}{n}=180-\frac{360}{n}+30.$
$-\frac{180}{n}=-\frac{360}{n}+30$
$\frac{180}{n}=30$
$n=6.$