# 多边形本文重定向自 多边形

## 分类

### 简单多边形

• 点在多边形内：决定一点是否在多边形内
• 多边形面积
• 将多边型切割成三角形

### 正多边形

• 面积：${\displaystyle A\ =\ {\frac {n}{2}}\,a\,r_{i}\ =\ {\frac {n}{2}}\,r_{u}^{2}\,\sin {\frac {2\pi }{n}}\ =\ {\frac {1}{4}}na^{2}\cot {\frac {180^{\circ }}{n}}}$
• 内切圆半径：${\displaystyle {\frac {a}{2}}\cot {\frac {180^{\circ }}{n}}}$
• 外接圆半径：${\displaystyle {\frac {a}{2\sin {\frac {180^{\circ }}{n}}}}}$

## 公式

### 面积

${\displaystyle A={\frac {1}{2}}\left({\begin{vmatrix}x_{1}&y_{1}\\x_{2}&y_{2}\end{vmatrix}}+{\begin{vmatrix}x_{2}&y_{2}\\x_{3}&y_{3}\end{vmatrix}}+\dots +{\begin{vmatrix}x_{n}&y_{n}\\x_{1}&y_{1}\end{vmatrix}}\right)}$

{\displaystyle {\begin{aligned}A={\frac {1}{2}}\{a_{1}[a_{2}\sin(\theta _{1})+a_{3}\sin(\theta _{1}+\theta _{2})+\cdots +a_{n-1}\sin(\theta _{1}+\theta _{2}+\cdots +\theta _{n-2})]\\{}+a_{2}[a_{3}\sin(\theta _{2})+a_{4}\sin(\theta _{2}+\theta _{3})+\cdots +a_{n-1}\sin(\theta _{2}+\cdots +\theta _{n-2})]\\{}+\cdots +a_{n-2}[a_{n-1}\sin(\theta _{n-2})]\}\end{aligned}}}

N边形S=${\displaystyle {\frac {\sum {(-1)^{k}mn\sin {\theta }}}{2}}}$这个代表N边形已知（N-1）个边的长度，而且知道其中任意两边的夹角，对于这两边${\displaystyle (-1)^{k}mn\sin {\theta }}$求和后的一半便是面积