四边形与多边形

考什么 · What it tests

需要先会 · Prerequisites

• 三角形面积与内角和 $180°$ / Triangle area and the $180°$ angle sum
• 平行、垂直的判定 / Recognizing parallel and perpendicular lines
• 割补法（面积与周长综合里学过） / Cut-and-paste (from the area-and-perimeter topic)

常见套路 · Common moves

1. 内角和一条公式：$n$ 边形内角和 $=(n-2)\times180°$，而外角和永远是 $360°$（与边数无关）——多边形角度题的总闸。

   One angle-sum formula: an $n$-gon's interior angles sum to $(n-2)\times180°$, while the exterior angles always total $360°$ regardless of $n$ — the master switch for polygon angle problems.
2. 割补成三角形：任何多边形从一个顶点连对角线，切成 (n-2) 个三角形，面积和角度都从三角形来。

   Cut into triangles: drawing diagonals from one vertex splits any polygon into (n-2) triangles, and both area and angles follow from them.
3. 梯形面积记「上底加下底」：$\displaystyle \frac{(a+b)h}{2}$，其中 $h$ 是两底之间的垂直距离，不是腰长。

   Trapezoid area is "top plus bottom": $\displaystyle \frac{(a+b)h}{2}$, where $h$ is the perpendicular distance between the parallel sides, not a leg.

易错点 · Common pitfalls

• 外角和以为随边数变：正多边形外角和恒为 $360°$，每个外角是 $360°/n$——别去乘 (n-2)。

  Thinking the exterior-angle sum grows with the sides: it is always $360°$, so each exterior angle is $360°/n$ — don't multiply by (n-2).
• 梯形的高错用成斜腰：面积公式里的 $h$ 必须是两底的垂直距离。

  Using a slanted leg as the height: the $h$ in the area formula must be the perpendicular distance between the two parallel sides.